(UNIT 3 - FUNCTIONS) Algebra 1 exam prep
direct variation
a function of the form y = kx where k is not zero
constant of variation
the value of k in a direct variation, y = kx
rate of change
how the dependent variable changes with respect to the independent variable
standard deviation
a measure of the distribution or spread of a set of numbers.
coordinates
an ordered pair, (x, y), that describes the location of a point in the coordinate plane
A line has a rise of 3 and a run of 12. What is the slope?
answer : 1/4 slope = rise/run = 3/12 = 1/4.
quadrant
region in the coordinate plane
Write function notation for "The abscissa increased by five is equal to the ordinate."
f(x) = x + 5.
Given: 2x + 3y = 6. What is the y-intercept?
(0, 2). y- intercept : Value of y on which x = 0. we know that x = 0 2 (0) + 3y = 6 0 + 3y = 6 3y = 6 - 0 3y/3 = 6/3 y = 2. y - intercept is 2 So, (0 , 2)
What is the next term for the given arithmetic sequence? -3, -2.25, -1.5, -0.75, ...
0
Which of the following represents x = 1/3y written in general form?
3 x - y = 0. x = 1/3y multiply both sides by the denominator. 3x = 1y or 3x = y change to general form 3x - y = 0.
Explain why the vertical line test is used to determine if a graph represents a function.
When a vertical line passes through the graph of a function, it will intersect the graph at only one point. More than one point on any given vertical line means the graph is not a function. This test is ideal and accurate for checking whether the graph is one of a function or not.
function
a relation in which each first coordinate is paired with exactly one second coordinate
translation
a shift or slide of a graph in the coordinate plane
slope of a parallel line
parallel lines have the same slope.
correlation coefficient of -1
tells you that the correlation is perfectly negative. That is, if one event occurs, the other event does not occur. For example, if the sun shines more, the days do not get colder.
mean
the average value of a set found by adding up the values in set and then dividing by the number of values in the set.
common difference
the fixed amount added on to get to the next term in an arithmetic sequence
x-axis
the horizontal axis in the coordinate plane
What is the equation of the line described below written in slope-intercept form? the line passing through point (2, 4), parallel to the line whose equation is y = x
y = x + 2. y = x or y = 1x. So, we know that the slope is 1. Parallel lines have the same slope, so now for the second equation, we already have the slope. y = mx + b (replace the values with the coordinates we have been given.) 4 = 1(2) + b 4 = 2 + b 4 - 2 = b 2 = b. so the equation is now y = x + 2.
What is the range of the function y = x^2?
y ≥ 0
Find the common difference for the sequence shown. 56, 49, 42, 35, ...
-7
parallel lines
lines that never intersect
explaining this '[1,3)'
the range is 1 to 3 (including 1 but not including 3) and then 7 to 9 (including both 7 and 9). Symbolically, we write this as [1,3) ∪ [7,9].
The graph of the function B is shown below. If B(x) = -1, then what is x?
answer : 2 b(x) represents y. So, another way to put this is y= -1, look on the graph for the point -1 on the y-axis, it connects with the point 2 on the x-axis
Determine the constant of variation for the direct variation given. (0, 0), (3, 12), (9, 36)
answer : 4 Okay, so choose any of the points given to us. (except (0,0) cause remember 'k' is the constant of variation and it can never be equal to 0.) I'll do both. y = kx. y = kx (3,12) (9, 36) 12 = 3k 36 = 9k 12/3 = 3k/3 36/9 = 9k/9 4 = k. 4 = k.
slope of a perpendicular line
the slope of a perpendicular line is the opposite of the reciprocal of the first line. Say the first line has a slope of 4, the second line will have a slope of -1/4.
If the ordinate is three more than twice the abscissa and x∈ {-1, 0, -1/3}, which of the following sets of ordered pairs satisfies the given conditions?
{(-1, 1), (0, 3) , (-1/3, 7/3)}. 'the ordinate is three more than twice the abscissa' so, y = 2x + 3. Choose any coordinate and replace them in this equation to determine which ordered pair satisfies the condition.
The graph of the function B is shown below. Find B(1)
1) Locate 1 on the x axis, it connects with the point 2 on the y-axis. Therefore b(1) = 2.
Describe the location of the point having the following coordinates. nonzero abscissa, negative ordinate
Quadrant III or Quadrant IV
Select the points that lie on the function h(x) = 3x^2.
- (1, 3) - (-1, 3) Replace X with one of the x-values, so h(x) = 3(1)^2 which equals to 3. Remember that h(x) represents y. So, h(x) = 3 is the same as y = 3. Among the options, look for the coordinates whose y values are 3.
The graph shown is a direct variation. What is the rate of change?
5. There's two ways of doing this, both simple! I'll show both. Method 1 : The point (10,50) meet. Direct Variation formula is y = kx 50 = 10k 50/10 = 10k/10 5 = k. (this method is preferable; it takes less time and uses direct variation, which is like.. the whole point of this muggle lesson.) Method 2 : Remember that the slope represents the rate of change. Choose two points that meet. (0,0) and (10,50). slope of a graph of 'm' = y2 - y1/x2 - x1 m = 50 - 0/ 10 - 0 m = 50/10 = 5/1 m = 5.
Describe the location of the point having the following coordinates. negative abscissa, positive ordinate
Quadrant II
In which quadrant is (1/2, -1.8)?
Quadrant IV
Extend the sequence given to the seventh term. 1.5, 1, 0.5, 0, ...
The common difference is -0.5: 1 - 1.5 = -0.5 0.5 - 1.0 = -0.5 0 - 0.5 = -0.5 The next three terms of the sequence are: 0 + (-0.5) = -0.5 -0.5 + (-0.5) = -1 -1 + (-0.5) = -1.5The seventh term of the sequence is -1.5.
Sarah is renting a car for her weekend trip to the mountains. The total cost of the rental, f(x), as it relates to the number of miles driven, x, is shown in the graph below. Write a function to represent this situation.
The function will be f(x) = 0.25x + 25. At 0 miles, the cost is 25, meaning 25 is the constant. So the function starts out as: f(x) = ? + 25. Between 0 and 40 miles, the cost rises by 10. Therefore, f(x) = (10/40)x + 25 f(x) = (1/4)x + 25 f(x) = 0.25x + 25
Using graph paper, determine the line described by the given point and slope. Click to show the correct graph below. (-2, 3) and 0
The graph will be a horizontal line with only the point (-2,3). The slope does not require us to move at all, therefore no other point will be made.
f-inverse
The inverse function of f is written f -1 and pronounced "f-inverse."
The scatter plot below shows the profit earned each month by a new company over the first year of operation. The owner writes a line of best fit equation, shown below, to model the relationship between profit earned and month. y=2,500x−2,500 Explain how you know that the line of best fit equation is appropriate. Be sure to mention both the slope and y-intercept in your response.
We are given with the best fit line y = 2,500x - 2,500 The y-intercept is -2500 and the slope is 2500 This line can be translated into the following statements: At the start of the operations, there is a negative profit of -2500. Each year, the profit increases by 2,500.
set
a collection or group of objects indicated by braces, { }
arithmetic sequence
a sequence in which a fixed amount is added on to get the next term
What is the value of k in the function f(x)=112−kx if f(−3)=121?
k = 3. Both the equations are equal to f(-3), so we must make them equal to each other. 112 - k(-3) = 121 112 + 3k = 121 3k = 121 - 112 3k = 9 3k/3 = 9/3 k = 3.
Using the given points and line, determine the slope of the line. (1, 2) and (2, 1)
slope = -1. (remember that rise represents the vertical change, y values. and run represents the horizontal change, x values.) rise = y2 - y1 = 1 - 2 = -1 run = x2 - x1 = 2 - 1 = 1 slope = rise/run slope = -1/1 slope = -1.
correlation coefficient of 1
tells you that the correlation is perfectly positive. That is, if one event occurs, the other event occurs, too. For example, if the sun shines more, the days get warmer.
y-axis
the vertical axis in the coordinate plane
Given: x + 2y = -6. Solve for y.
y = -x - 6/2. x + 2y = -6 2y = -6 - x y = -6 - x/2 or -x - 6/2
Write the equation 2x - 3y = 6 in slope-intercept form.
y = 2/3x - 2. write the equation in slope-intercept form, so, -3y = -2x + 6 3y = 2x - 6 divide both sides by 3. 3y/3 = 2x - 6/3 y = 2/3x - 2.
slope-intercept form
y = mx + b form of a linear equation
The cost to mail a package is $7 for the first 2 pounds and 30 cents for each additional ounce. Which of the following functions represents the cost to mail a package if x is the number of ounces over 2 pounds?
ƒ( x ) = 7 + 0.3 x. y or f(x) is the total cost, 7 is the constant, and x is the number of ounces over 2 pounds (you'd have to multiply x by 0.3 to find the cost for all the pounds ordered).
linear regression
a statistical tool to find a line of best fit and the correlation coefficient.
Solve for y in terms of x. 2/3y - 4 = x
y = 3/2x + 6. 2/3y - 4 = x 2/3x - 4 = y 2/3x (3) - 4(3) = (3)y 2x - 12 = 3y 2x/2 = 3y + 12/2 x = 3/2y + 6 y = 3/2x + 6.
Write the equation for 5x + 2y = 3 in slope-intercept form.
y = -5/2x + 3/2. write the equation in slope-intercept form, 2y = -5x + 3 divide both sides by 2. 2y/2 = -5x + 3/2 y = -5/2x + 3/2
Which of the following functions is a direct variation?
f(x) = 2x.
correlation coefficient
how strong or weak the correlation is between two events.
On a separate piece of graph paper, graph y = -|x| + 1; then click on the graph until the correct one appears.
the upside down v at (0,1). We already know that v will be upside down when we are given y = -|x|, so when +1 is added, it calls for a vertical change, meaning it should be moved 1 unit up, it'll go from (0,0) to (0,1)
dependent variable
the variable representing the second elements of the ordered pairs in a function; the outputs
For the graph y = 4 find the slope of a line that is perpendicular to it.
undefined
correlation
when one event can be used to predict another.
causation
when one event causes another.
linear equation
an open sentence whose graph is a line; the general form for such an equation is Ax + By + C = 0
Find the tenth term of the sequence 0, 5, 10, 15, 20, ....
term number: n = 10 (we are finding the tenth term) first term: a = 0 common difference: d = 5 A(n) = a + (n - 1)d A(10) = 0 + (10 - 1)(5) A(10) = 0 + (9)(5) A(10) = 0 + 45 A(10) = 45 The tenth term of this sequence is 45.
absolute value
the distance that a number is from zero on the number line; always positive
abscissa
the first number (or x value) of an ordered pair
origin
the point of intersection, (0, 0), of the axes in the coordinate plane
y-intercept
the point where the line crosses the y-axis
slope
the rate of change of a line; change in y over change in x; rise over run
independent variable
the variable representing the first elements of the ordered pairs in a function; the inputs
residual
the difference between the value predicted by an approximating function (like a line) and the observed value.
ordinate
the second number (or y-value) of an ordered pair
domain
the set of first coordinates in a relation
Which of the following describes how to translate the graph y = |x| to obtain the graph of y = |x + 7|?
7 units left. if you have (y = |x + c|) , it would shift c units to the left.
range
the set of second coordinates in a relation
Different Ways to Express Slope
- slope = rise (or drop)/run - vertical change/ horizontal change - Δy/Δx - y2 - y1/x2 - x1
Inverse function
A function f -1 that switches the input and output of the original function f. So, if f( x) = y then f -1( y) = x.
Whenever you sign a lease for an apartment, you typically have to pay a security deposit before you even move in; this is in case something happens while you care occupying the apartment and repairs have to be made. One apartment building has apartments that rent for $750 a month and a security deposit of $300. Write an equation to represent the situation described.
F( x) = 750 x + 300. Let x be the amount of months for which rent is paid (so let's say you decide to pay 3 months' worth of rent, x would be 3), and y be the total cost. $300 dollars is the constant. To find the total cost, you have to multiply the rent (750) by the amount of months you paid for so, 750x. Now you have the total price rent will cost you, but remember that there is an additional 300 dollars for the security deposit, so you'd have to add 750x to 300 to get the total cost of everything. so, f(x) = 750x + 300
perpendicular lines
lines that intersect to form right (90 degree) angles
Click on the graph to show the one that represents the following problem. The ordinate is twice the abscissa and x∈{-1, 0, 1}.
the graph with the points, (-1,2) (0,0) (1,2). 'the ordinate is twice the abscissa' so we already know that y = 2x. take the x values we've been given {-1, 0, 1} and replace them in the equation y = 2x to find the y-coordinates!
Write the equation of the line that has an x-intercept of -3 and passes through the point (-3, 7).
x = -3. We have the points (-3, 0) and (-3,7). The slope is undefined and there is no y-intercept, so, the 'm' and 'b' part of the equation are out. That leaves us with y = x. We know that y = 0 so, 0 = x. Now we are basically just left with x. We know that the x-intercept is -3 so, x = -3.
For the graph y = 12 find the slope of a line that is perpendicular to it and the slope of a line parallel to it. Explain your answer with two or more sentences.
A graph of the line y = 12 has a slope equal to zero since it lies flat crossing the y-axis at point 12. The slope of the line perpendicular should be infinity since the line perpendicular straight up crossing the x-axis. The slope of the line parallel should be zero too.
The graph of a line passes through the two points (-2, 1) and (2, 1). What is the equation of the line written in general form?
y - 1 = 0. The slope is equal to 0. (calculate this. Remember that y2-y1/x2-x1 is a way of finding the slope with 2 coordinates.) If the slope is 0, then 0x would equal to 0 no matter what the value of x was so the 'mx' part of the slope-intercept equation is taken out. we are left with y = 1. (because both y coordinates are equal to 1 and there is no y-intercept.) Change y = 1 to general form, it'll be y - 1 = 0.
You are 495 miles from home and are driving toward home at an average speed of 45 mph. How long will it take to drive home if you maintain the same speed? (Hint: When you reach home, the y-value is zero.)
Distance=speed x time 495=45 x t t=11
For the graph x = 12 find the slope of a line that is perpendicular to it and the slope of a line parallel to it. Explain your answer with two or more sentences
The slope of a line parallel to the y -axis has a slope that is undefined. x=12 will have undefined slope. Any line perpendicular to x=12 will become parallel to x axis and will have slope =0.
The slope of a line is -1/3, and the y-intercept is 10/3. What is the equation of the line written in general form?
x + 3y - 10 = 0. First, write it in the slope-intercept form y = mx + b where m = slope and b = y-intercept. so y = -1/3x + 10/3, this is our slope-intercept form, now we change it to general form. General form is ax + by + c = 0. Where ax must always have a positive coefficient. Multiply both sides by the denominator 3, it becomes 3y = -1x + 10, take all of them to the left side of the equation and make it equal to 0, so now it's 1x + 3y - 10 or x + 3y - 10! That is our equation in general form. just some more explanation; let's say that we end up with a general form equation of -2x - 3y + 6, this would be wrong. Because remember that the coefficient of the first term in a general form equation cannot be negative. To make it positive, multiply all the terms by -1, so it becomes 2x + 3y - 6.
The graph of a linear function h contains the points given in the table below. Find h^-1.
h^-1( x) = - 1/2x. Choose any two points on the table given to us and find the slope. The slope is -2/1, the inverse of -2/1 is -1/2. or, you could choose any two points on the table and switch them (for example (-2,4) will become (4, -2)), then find the slope which will be -1/2. Now, find the y-intercept using the slope of the inverse function and any of the points on the table, i chose (-2,4), remember i can't find the inverse y-intercept with the points of the original function so switch (-2,4) so it becomes (4,-2). -1/2 = y - (-2)/x -4 cross multiply. 2y + 4 = - x + 4 2y + x = 4 - 4 2y + x = 0 2y = - x 2y/2 = -x/2 y = -x/2 or -1/2x (you can use the slope of the original equation along with one of the original points to find the y-intercept but it would take longer.)
What is the equation of the following line written in slope-intercept form?
y = -7x - 11 on the graph, we can see the points (-2,3) and (-1,-4). We are being asked to write this in slope-intercept form y = mx + b where m is the slope and b is the y-intercept. find the slope y2-y1/x2-x1. Then use the slope (which will be -7/1 = -7.) and choose any of the two points given to us and replace them in the slope-intercept equation. I'll be using (-2,3). y = mx + b 3 = -7(-2) + b 3 = 14 + b 3 - 14 = b b = - 11
A sporting goods store is having a 15% off sale on all items. Which functions can be used to find the sale price of an item that has an original price of x? You may choose more than one correct answer.
y = 0.85 x (Taking 15% off is the same thing as only charging 85%) f( x) = x - 0.15 x Sale = Original - .15(Original)
Which equation describes a relationship such that x = 0 must be excluded from the domain?
y = 2/x. Replace x in the equation with 0. That would mean y = 2/0 which is a math error. Therefore x cannot equal to 0 in this equation. (remember that domain is the x values and range is the y values)
find the inverse function of the equation f(x) = 1/3x - 4.
y = 3x + 12. or f(x) = 3x + 12. remember that f(x) is just another way of writing y. so y = 1/3x - 4. switch x and y. so now, x = 1/3y - 4 make 1/3y the subject by moving -4 to the left side of the equation. so, x + 4 = 1/3y. multiply both sides by the denominator. 3(x+4) = 1/3y (3) 3x + 12 = y. or f(x) = 3x + 12.
Write the following equation in the general form Ax + By + C = 0. 2x + y = 6
2x + y - 6 = 0
Which of the following represents "The ordinate is twice the abscissa"?
2x = y
Find the 20th term of the following sequence. -6, -4, -2, 0, ...
32
Find the range of the following piecewise function.
[−5,−2)∪[3,13)
A function is _____ a relation.
always
Describe the location of the point having the following coordinates. positive abscissa, positive ordinate
Quadrant I
In which quadrant is (-1/2, -1.8)?
Quadrant III
Find x if f(x) = 2x + 7 and f(x) = -1.
To find x when f(x) = 2x + 7 and f(x) = -1, we must make these equations equal to each other. This is because they are both equal to f(x). We can then use algebra to solve for x. 2x + 7 = -1 2x = -8 x = -4
Select all of the following that are ordered pairs of the given function. ƒ(x) = 3 - 2x
- (2, -1) - (0, 3) - (-1, 5) It's simple. Take the X value from the coordinate and replace it in the equation, then whatever answer you get should be equal to the Y value from that same coordinate. For e.g: (2, -1) ƒ(x) = 3 - 2x replace x with the x value from the chosen coordinate f(2) = 3 - 2(2) f(2) = -1 f(2) represents y, so, y = -1. Meaning (2,-1) is an ordered pair of the function f(x) = 3 - 2x
A line has a slope of 1/2 and a run = 50. What is the rise?
answer : 25 slope = rise/run 1/2 = x/50 1/2 /1 = x/50 cross multiply 50 x 1/2 = x 25 = x.
The graph below shows the total cost of an internet subscription, C, as it relates to the number of internet hours used, h. How much does the company charge per hour?
answer: $5 We are being asked to find the rate per hour, since the slope of a graph represents the rate at which the value increases or decreases, we can solve this by finding the slope of the graph. Choose any two points that meet on the graph. I chose (0, 35) an (3, 50). slope of a graph or 'm' = y2 - y1/x2 - x1 m = 50 - 35/3 - 0 m = 15/3 m = 5
If f(x) is a function and f(1) = 5, then which of the following could not be true?
f(1) = 1 could not be true. remember that f(1) is representing y. So, f(1) = 5 is the same as y = 5. So now we know that x = 1 and y = 5. A function cannot have any repeating x coordinates. We have been given the following options : f(5) = 5 which is (5,5) f(2) = 1 which is (2,1) f(1) = 1 which is (1,1) Look at the X coordinates for each option; 5, 2 and 1. In the question the function is f(1) = 5 which is 1,5. The reason we say f(1) = 1 cannot be apart of this function is because 1 would appear twice among the X coordinates, and remember that a function cannot have an X coordinate that appears more than once. (it's okay for y coordinates to appear more than once in a function)
Susan is looking into a new cell phone carrier. She researches her options and finds the two companies which have the best reception in her neighborhood offer very different monthly plans. Write a function rule which represents each company's monthly fees. A) Company A offers unlimited minutes and texts for a flat fee of $50 per month and charges $1.50 for each GB of data used per month. B) Company B offers unlimited minutes and texts for a flat fee of $35 per month and charges $5.00 for each GB of data used per month.
Let x represent the number of GB of data used per month . Let f(x) be the function that represents the total cost . A) For Company A it is given :Company A offers unlimited minutes and texts for a flat fee of $50 per month and charges $1.50 for each GB of data used per month. Charge for x GB data used = 1.50x. Total cost f(x)= 1.50x + 50 B) Given :Company B offers unlimited minutes and texts for a flat fee of $35 per month and charges $5.00 for each GB of data used per month. Cost of x GB data = 5x Total cost f(x) = 5x + 35.
Using graph paper, determine the line described by the given point and slope. Click to show the correct graph below. (0, 0) and 2/3
Look for the graph that has the points (0,0) (3,2). This is simpler than it may look. Δy/Δx = 2/3 so Δy = 2 and Δx = 3 , meaning 'rise' (vertically) 2 units then 'run' (horizontally) 3 units from (0,0). the second set of points will be (3,2)
An equation is written to represent the relationship between the temperature in Alaska during a snow storm, y, as it relates to the time in hours, x, since the storm started. A graph of the equation is created. Which quadrants of a coordinate grid should be used to display this data?
Quadrant 1 and 4 only. Y = temperature X = time Time cannot be negative, so for X, only quadrant 1 and 4 can be used. Temperature can be both positive and negative. Quadrant I = +x,+y this works because x can only be positive and y can be either positive or negative Quadrant II = -x,+y this does not work because x cannot be negative Quadrant III = -x,-y this does not work because x cannot be negative Quadrant IV = +x,-y this works because x can only be positive and y can be either positive or negative
Describe the location of the point having the following coordinates. abscissa and ordinate have the same sign
Quadrant I or Quadrant III
Which of the following represents "Twice the abscissa increased by three times the ordinate is ten"?
2x + 3y = 10
quadratic function
a function that contains x^2 as the highest degree variable
The balance of a loan is $2,570 in January, and the monthly payment is $125.50. The relationship between the loan balance, y, and the number of monthly payments made since January, x, can be represented by the equation y = 2,570 − 125.50x. In what months does the loan balance, y, meet the condition, $1,600 < y < $2,000?
answer : June, July and August. To solve this, you have to break it down and understand what the question is asking. y = relationship between loan balance x = number of monthly payments since January. the equation is y = 2,570 - 125.50x the question asks 'In what months does the loan balance, y, meet the condition, $1,600 < y < $2,000?' $1,600 < y < $2,000 put in words states that 'the loan balance is more than $1,600 but less than $2,000' so basically, in what months was the loan balance more than $1,600 but less than $2,000. We know that January's loan balance is $2,570. To find the loan balance of the previous months : start by subtracting $125.50 from January's loan balance to find the loan balances of the previous months 2,570 - 125.50 = 2,444.5 (this is December's loan balance, it does not meet the condition.) 2,444.5 - 125.50 = 2,319 (November, does not meet condition) 2,319 - 125.50 = 2,193.5 (October, does not meet condition) 2,193.5 - 125.50 = 2,068 (September, does not meet condition) 2,068 - 125.50 = 1,942.5 (AUGUST MEETS THE CONDITION) 1,942.5 - 125.50 = 1,817 (JULY MEETS THE CONDITION) 1,817 - 125.50 = 1,691.5 (JUNE MEETS THE CONDITION) Now we know any month before June cannot meet the condition because their loan balance will be less than $1,600
Describe the location of the point having the following coordinates. negative abscissa, zero ordinate
between Quadrant II and Quadrant III
Describe the location of the point having the following coordinates. zero abscissa, negative ordinate
between Quadrant III and Quadrant IV
The table shown below represents a function. Which of the following values could not be used to complete the table? the values shown on the table : x values = -10, -25, -5, ? (that represents the missing x value) y values = 5, 10, 15, 20
-5 cannot be used to complete the table, this is because a function cannot have any repeating x coordinates.
Which of the following describes how to translate the graph y = |x| to obtain the graph of y = |x| - 4?
4 units down. If you subtract a constant outside the absolute value symbol, it shifts the graph downward. (y = |x| - c)
Which equation could represent "the area of a square as a function of a side"?
A(s) = s^2 the area of a square is a = s^2 when converted to a function notation, the equation becomes A(s) = s^2
Tomar's paycheck varies directly with the hours he works. When Tomar works 5 hours, he gets paid $60.00. How much does Tomar make in 8 hours?
Answer : $96 First, find the constant of variation. $60.00 = 5 hours y = kx 60 = 5k 60/5 = 5k/5 k = 12. Amount Tomar makes in 8 hours will be : y = amount tomar makes in an hour y = kx y = 12 x 8 y = $96
Find the range of the following piecewise function.
[0,11). For this, you'll need the lowest and highest range values. In the first function, the lowest range (y-coordinates) value is 0. In the second function, the highest range value is 11. (but remember, 11 isn't included in this (meaning instead of the function being x equals to or is less than six, it's x is less than 6, so it isn't included), so instead of the answer being [0.11], it'll be [0,11) because six's matching coordinate is eleven, and remember, they aren't included.
piecewise-defined function
a function that has different rules for different parts of its domain
linear function
a function which has one or two variables with no powers, this functions' graph will always be a straight line
sequence
a set of numbers that follow a pattern, with a specific first number
relation
a set of ordered pairs
solution
a value or values of the variable that make an algebraic sentence true
coordinate plane
a way of locating points in a plane that consists of a horizontal and a vertical number line intersecting at the zeros
Eliza started her savings account with $100. Each month she deposits $25 into her account. Determine the average rate of change in Eliza's account from the 2nd month to the 10th month. Show all work for full credit.
since she started with $100, then that would be a constant. So the function would start off with f(x)=?+100. since she is depositing $25 per month, then the 'x' variable would be per month. So the function would be f(x)=25x+100.
Given: 5x - 6y = 0. What is the x-intercept?
(0,0). 5x - 6y = 0. 6y = -5x + 0 we know that x = 0. 6y = -5(0) + 0 6y = 0 6y/6 = 0/6 y = 0.
Given that D(x) = 2x, select all of the following that are true statements.
- D( x) is a rule for the set of points (5, 10), (6, 12) and (-2, -4). - D( x) is a direct variation. - D( x) is a function. D( x) is a rule for the set of points (5, 10), (6, 12) and (-2, -4). → this is true because when you place the x values in this function into the formula, the y-value is the same as the one in the coordinate from which the x-value came from. For example, let's use the given coordinate (5,10). d(x) = 2x → d(5) = 2(5) = 10. d(5) = 10 is the same as y = 10. D( x) is a direct variation. → Direct variation formula is y = kx. In this question the equation is a direct variation where 2 represents k as the constant. D( x) is a function. → Yes, D (x) is a function of x.
Select all of the following that are a direct variation.
- Tickets for rides at the fair are $1 each. - Songs are 99 cents each to download.
stepwise function
a special type of a piecewise-defined function in which each piece is a linear function
Which of the following functions has a graph that is a line?
f(x) = x. This given function is the same as the expression y = x (because remember that f(x) represents y). So lets say the y coordinates equal to 1,2,3, and 5 respectfully, that would mean the x coordinates also equal to 1,2,3, and 5. so (1,1) (2,2) (3,3)... When plotted on a graph and the points are connected, it'll be a straight line.
If the points (0, 0), (-1, 1), and (-2, 2) lie on the graph of function h, then which of the following rules could represent the function?
h( x) = | x|. Take (-2,2) for example, h(x) = |-2| h(x) = 2 y = 2.
If the graph of y = |x| is translated so that the point (1, 1) is moved to (1, 4), what is the equation of the new graph?
y = |x| + 3. We see that the y-value has changed from 1 to 4, that means the difference of 3. So there has been a vertical change. We know that when y = |x|, if a constant is added outside the absolute value of the symbol, then it shifts upwards by 3. hence, y = |x| + 3.
The hourly wage increase each employee receives each year depends on their number of years of service. Every three years of service means an increase of $0.50 per hour. So, employees who have been with the company for less than three years can expect to receive an increase of $0.50 per hour. Employees who have been with the company for at least three years, but fewer than six years can expect an increase of $1.00. Employees who have been with the company for at six years, but fewer than nine years, receive an increase of $1.50 per hour. And, employees of at least nine years, but fewer than twelve years receive an increase of $2.00.
years = 1 wage increase 0.50 years = 3 wage increase 1.00 years = 6 wage increase 1.50 years = 9 wage increase 2.00 years = 12 wage increase 2.50
If the ordinate equals the abscissa and x∈{-3, 0, 12}, which of the following sets of ordered pairs satisfies the given conditions?
{(-3, -3), (0, 0), (12, 12)}. Ordinate is the y coordinates and abscissa is the x coordinates. so if ordinate equals equals the abscissa then y = x meaning the y and x coordinates are the same.
Which of the relations has a domain of {-5, 0, 5}?
{(-5, 5), (0, 5), (5,5)} domain is the x coordinates and in this relation the x coordinates are -5, 0 and 5.
