Math - Introduction to Functions

What is the "b" used for in f(x) = ax^2 + bx + c

(general form) "b" is used to obtain the COORDINATES of the vertex (highest or lowest point) of f(x). It is also used to obtain the x-intercept of f(x)!

If f(x) = 3x^2 + 4x - 20, what is the y-intercept?

(general form) Remember that the y-intercept MUST BE A POINT. Since "c" is the y-intercept, it is: (0, -20)

What must we remember in f(x) = a(x-h)^2 + k

(standard form) The "h" is ALWAYS the "OPPOSITE" of what we see!!!

How do we obtain the vertex of f(x) if f(x) is in the general form?

*This only works for the general form* We use the following formula: V --> (-b/2a , 4ac - b^2/4a) (x , y)

What two methods can be used to get the x-intercepts of the quadratic in the standard form?

1) Algebraic method 2) Using a formula

What do we need before sketching a quadratic?

1) Concavity 2) Vertex 3) x-intercepts *You must include all three in the graph

How would we find the rule from a graph?

1) First determine what sort of function it is looking at the line. For example, let's say a graph is an increasing direct function. 2) Find the rule based on the function. A direct function's rule is: f(x) = ax 3) Find "a", the slope! y2 - y1 over x2 - x1 (read y2, y1, x2, x1 from left to right). Let's say that when we calculate "a", we get 4/3. (we don't want a decimal unless it's a nice decimal) 4) So there we have it! f(x) = 4/3x Why didn't we solve for x? Because we already know what "x" is because we were given the points.

What are the three possible rules of a quadratic function? Name it and give it its rule.

1) General Form: f(x) = ax^2 + bx + c 2) Standard Form: f(x) = a(x-h)^2 + k 3) Zero form OR factored form: f(x) = (x-x1)(x-x2)

Outline the steps for finding the rule of a partial function.

1) Get the slope (a = y2 - y1 over x2 - x1) 2) Plug the result in: f(x) = ax + b 3) Pick a point (don't forget to indicate which point you're choosing), replace x & f(x) (which means y) and solve for "b" 4) Give your final answer as (fx) = ax+b (plug in what you found for "a" and "b"

What is the slope of f(x) = 12 - 4x?

1) Reorder it: f(x) = -4x + 12 2) Answer: -4 Remember the rule? f(x) = ax + b "a" is the slope.

Outline the steps of using the algebraic method to get the x-intercepts of the quadratic in the standard form.

1) Replace f(x)/y by 0 2) Solve for x!

Use the algebraic method to get the x-intercepts of f(x) = -2(x+1)^2 +32

1) Replace f(x)/y by 0: 0 = -2(x+1)^2 + 32 -32 = -2(x+1)^2 Opposite of squared is square root. But before doing that, let's actually get a number that's rootable (furthermore, we cannot square root a negative number!): -32/-2 = (x+1)^2 (square root)16 = (x+1) 4 = x+1 OR -4 = x+1 (you might prefer to use a formula for this reason as it could be confusing) -1 +/- 4 = x Two options: (3, 0) AND (-5, 0)

What must we remember when graphing?

1) Scale according to numbers. 2) Scale by cm (on ruler) otherwise it won't be accurate. 3) Connect the dots, but go a bit past the dot. 4) Use the vertical line test! (if asked whether it is a function) 5) Title the graph: y vs. x *Do not skip lines when scaling! USE the cm on ruler. *NEVER graph a decimal nor a fraction. That means you need to keep looking for another x to get a whole number for y.

4 ways in which functions can be represented in

1) Sets of elements 2) Graph 3) Sets of points 4) Rule/Equation

Sets of elements 1) How many sets of elements will be given? 2) Describe the association. 3) When is it a function and when is it a relation?

1) Two 2) The elements of the source set will be associated/matched to the elements of the target set. 3) Relation; When an element of the source set is associated/matched to more than one element of the target set Function; When an element of the source set is associated/matched to one only one element of the target set

What is the equation of the axis of symmetry for f(x) = x^2 + 10x + 21? Find the vertex with this as well. What is the y-intercept?

1) Use the formula: x = -b/2a 2) Simply plug it in: x = -10/2(1) x = -10/2 x = -5 If you wanted to find the vertex... (remember it is a POINT). f(x) = x^2 + 10x + 21 We know that x = -5, so plug that in to get your f(x): f(x) = (-5)^2 + 10(-5) + 21 f(x) = -4 The vertex is therefore (-5, -4) (x , y) An alternate to this is just using the vertex formula. Now for the y-intercept... The "c" is the y-intercept, so the y-int is (0 , 21)

How would we graph f(x) = 12x?

1) We would construct a table of values, so replace "x" by 0, whatever you calculate is your y (it is 0), then you could replace "x" by 0, and we calculate y = 12. *Two points is enough! 2) Head on straight to graphing! Here you could tell what kind of function it is: it is an increasing direct function!

What is needed when finding the slope of a direct function?

2 points!

Describe what a decreasing partial function looks like. What sign is "a"?

A DECREASING straight line that does NOT go through (0,0)! "a" is negative. Why is it negative? TRICK: decreasing = negative

What does the graph of the quadratic function look like?

A curve that looks like a "U"

Define function.

A one-to-one relation between the elements of a SOURCE SET and the elements of a TARGET SET.

What is the difference between a sketch and graphing?

A sketch is an approximate visual. Graphing requires a table of values and plotting.

What does the graph of a direct function look like?

A straight line that always passes through (0,0)

What does the graph of a partial function look like?

A straight line that does NOT go through (0,0)! TRICK: Think of it as "partial," meaning, it does not go through the origin point, but the direct function does.

What does IR mean?

All real #'s.

What point is the y-intercept of a constant function?

Always the point (0,C)

Describe what an increasing partial function looks like. What sign is "a"?

An INCREASING straight line that does NOT go through (0,0)! "a" is positive. Why is it positive? TRICK: increasing = positive

What does the graph of the constant function look like?

An horizontal line passing through y = c. Basically, a horizontal line passing through the origin point, (0,C)

(0,0) -- is that a y-intercept or an x-intercept?

Both! (applies only for origin point, (0,0)

What does a closed and an open circle mean?

Closed = Included Open = Excluded

What trick can we associate with a concave up parabola and a concave down parabola?

Concave up = smiling Concave down = frowning

What are the two types of parabolas?

Concave up parabola and concave down parabola.

What are the types of functions?

Constant, direct, partial, quadratic and greatest integer (AKA step function)

What is the source set called?

DOMAIN of the function

Abbreviation of the domain of a function

Dom f

What are properties of a function?

Domain, range, intercepts, variations, signs, and extremes.

When dealing with signs of a function, are the x-intercepts included or excluded?

Excluded because they are positive nor negative.

Are critical points included or excluded from our intervals of increase or decrease?

Excluded.

How do we read the extremes of a function and how do we give our answer?

From bottom to top off of the y-axis, but our answer is given as a point.

What is the highest and lowest point called?

Highest point: Maximum Lowest point: Minimum

What is the y-intercept of f(x) = 2?

I can match the structure of the function to that of the rule of the constant function: f(x) = C And since y=C, the y-intercept is (0,2)

How do we know when x-intercepts exist in the standard form of the quadratic? (Will be tricked)

If "a" and "k" have opposite signs!!!

f(x) = a(x-h)^2 + k What is "a" used to determine?

If f(x) is concave up or down! Remember that if "a" is positive, it is concave up, and if "a" is negative, it is concave down.

Describe what the vertical line test consists of.

In drawing a vertical line passing through the "curve."

What are two types of graphs for a direct function?

Increasing direct and decreasing direct.

What are two types of graphs for a partial function?

Increasing partial and decreasing partial.

In a concave down parabola, what is the vertex?

Maximum (highest point)

In a parabola, the vertex is either a ____________ or ____________.

Maximum or minimum.

In a concave up parabola, what is the vertex?

Minimum (lowest point)

What is the vertex of f(x) = -4x^2 - 16 and is it a maximum or a minimum?

Normally, the general form would look like: f(x) = ax^2 + bx + c But since we do not have bx, b is just = 0. V --> (-b/2a , 4ac - b^2/4a) V --> (-0/2(-4) , 4(-4)(-16) - (0)^2/4(-4) V = (0 , -16) <-- that is your VERTEX! To determine whether it is a maximum or minimum, it would help to know the CONCAVITY. a = -4 (concave down) Or you could just plot the vertex down, but you wouldn't know whether it is a maximum or minimum without its concavity. Draw it out (it should look like a frown), and you will see that the VERTEX is a maximum, even though it's below the x-axis (still a maximum)

Using sets of points -- When does a set of points represent a function?

ONLY if the x-coordinates do NOT repeat in the x-coordinate position.

When does a graphed curve represent a function?

Only if the vertical line test is "passed."

f(x) = ax^2 + bx + c (general form), what are a, b and c called?

Parameters.

What are the signs of a function?

Positive & negative intervals of a function.

What is the target set called?

RANGE of the function

Abbreviation of the range of a function

Ran f

How would we find the y-intercept of the quadratic in the standard form?

Replace x by 0!

What does "a" represent in the rule of the direct function?

Rule of a direct function: f(x) = ax "a" is the slope of the function! It could also be named rate of change or variance.

What does the "a" in the general form give?

Rule: f(x) = ax^2 + bx + c "a" gives the concavity (up or down)

In the rule of a general form (quadratic), what common misconceptions are possible?

Rule: f(x) = ax^2 + bx + c "a" is NOT the slope (because it is a curve) "b" is NOT the initial value

Does x^2 + y^2 = 1 represent a function?

Solution: 1) Construct a table of values. Pick values for "x" and plug them in x^2 + y^2 = 1. 2) So we chose to replace x by 0, which is the easiest, so the x value is 0 and we were represented with two options: +/- 1. (this is not a function therefore) We then replace x by one. y = 0, so that's one option. We then replace x by 2. We cannot square root here, so 2 as an x-value does not work (because we cannot square root a negative!) We could keep on going... but you get the idea. 3) Construct a graph! Use the table of values that you constructed and plot your lines. Then pass the vertical line test... and it does not work. So, NO, IT IS NOT A FUNCTION.

How would I enter it on a calculator: x = -b +/- (square root): b^2 - 4ac / 2a

Square root AFTER you get delta (b^2 - 4ac). Or put brackets: (square root) (b^2 - 4ac)

Decreasing Direct Function -- what sign is "a"?

Straight line decreasing and that passes through (0,0): "a" is negative!

Increasing Direct Function -- what sign is "a"?

Straight line increasing and that passes through (0,0): "a" is positive!

a = y2 - y1 over x2 - x1 can only be used for what?

Straight lines (not for curves!)

What is a parabola?

The curve that represents a quadratic function.

What are the extremes of a function?

The highest and lowest points of a function.

What is the y-intercept also called?

The initial value of the function.

What does the "b" represent in a partial function?

The initial value or y-intercept.

What are the variations of a function?

The intervals of increase and decrease of the function.

What is a critical point?

The point where a change occurs (down to up or up to down)

What are the positive intervals of a function?

The portion of the graph that lies ABOVE the x-axis. (We read this interval from left to right off of the x-axis).

What are the negative intervals of a function?

The portion of the graph that lies BELOW the x-axis. (We read this interval from left to right off of the x-axis).

What are the intervals of decrease?

The portions of the function that seem to be MOVING DOWN when we read from left to right (off of the X-AXIS).

What are the intervals of increase?

The portions of the function that seem to be MOVING UP when we read from left to right (off of the X-AXIS).

What is the domain of a function?

The set of all x-values that define a function.

What is the range of a function?

The set of all y-values that define a function.

What does the "a" represent in a partial function?

The slope.

What is the x-intercept of a function?

The value of x when y is replaced by zero. It is, therefore, the point (x,0), and it is located on the x-axis.

What is the y-intercept of a function?

The value of y when x is replaced by zero. It is, therefore, the point (0,y) and it is located on the y-axis.

Define axis of symmetry of a quadratic function.

The vertical line that "cuts" the parabola into 2 EQUAL parts!

f(x) = a(x-h)^2 + k What are "h" and "k"?

The x & y coordinates of the VERTEX! (no work is therefore needed to find out the vertex)

Where does the axis of symmetry pass through?

The x-coordinate of the vertex.

What is the "c" of f(x) = ax^2 + bx + c

The y-intercept of the function (AKA f(x)!

What is the x-intercept also called?

The zero OR the root of the function

What are the x-intercepts of f(x) = x^2 + 6x + 5

There will always be TWO OPTIONS. Keep that in mind because your final answer will be two points. 1) Use the formula: x = -b +/- (square root): b^2 - 4ac / 2a 2) Plug everything in: x = -6 +/- (square root): 6^2 - 4(1)(5) / 2(1) For the calculator, do it in steps: x = -6 +/- (square root): 16 / 2 (1) x = -2 / 2 AND -10 / 2 x = (-1, 0) AND (-5, 0)

What are parameters?

Values that affect the shape of the parabola!

What property of function does a critical point pertain to?

Variations of a function.

What is the y-intercept of f(x) = -3(x+1)^2 - 6?

We know that it is in standard form. To get the y-intercept, all we need to do is replace x by 0: f(x) = -3(0+1)^2 - 6 (enter this in calculator and you get): f(x) = -9 But of course the y-intercept must be a point so the answer is: (0, -9)

What are the x-intercepts of f(x) = 2(x-4)^2 - 32?

We know that it is in standard form. f(x) = a(x-h)^2 + k Therefore, we use the formula to find the x-intercepts: x = h +/- (square root): -k/a x = 4 +/- (square root): 32/2 x = TWO OPTIONS: (8 , 0) and (0 , 0)

If f(x) = -2x^2 + 3x + 8, is the function concave up or down?

We look at "a" Since "a" is a negative, it is concave down. Trick: negative = down (mood)

When is the concavity up or down?

We look at the "a" in Rule: f(x) = ax^2 + bx + c If the "a" is positive (think happy), the parabola is concave up! If the "a" is negative (think sad), the parabola is concave down!

What must we remember when giving an interval, in general?

We must exclude the infinity sign and always put the sign, even when it's a positive.

Using a rule/equation -- What must we do when given a rule/equation?

We must graph the equation and then use the vertical line test.

How do we obtain the range of a function?

We must scan the graph from BOTTOM to TOP and read the start and end values off of the Y-axis.

How do we obtain the domain of a function?

We must scan the graph from left to right and read the start and end values off of the x-axis.

How do we know which is y2, y1, x2, and x1 when finding the slope?

We read the graph from left to right.

What is the y-intercept of y=2(x+4)^2 - 8?

We replace the x by 0 and just calculate for y! The answer is therefore (0,24) (0,y). *When we cannot square root a negative number, the I.V. or y-intercept DOES NOT EXIST!

What is the x-intercept of y=2x-8?

We replace the y by zero and solve for x. The answer: (4,0) (x,y)

How would we get the x-intercept of the quadratic in the general form?

We use the quadratic formula to get the x-intercepts!: x = -b +/- (square root): b^2 - 4ac / 2a

How would we find the axis of symmetry of the quadratic in the standard form?

We would use the formula: x = h

What are the intercepts of a function?

When a function has x and y intercepts.

When using the vertical line test, when would it represent a function?

When the curve is touched only ONCE by the line, the curve represents a function.

When asked for the zero of a function, do we need to put our answer as a point?

YES!

When asked for the extremes of a function, do we need to put our answers as a point?

YES! (even though we look at the y-axis to find our max. and min.)

Does S={(1,2), (2,3), (4,6), (5,8)} represent a function?

Yes! None of the x-values repeat.

Can a graphed curve also be a straight line?

Yes.

How do we find the slope "a" in a direct function?

a = y2-y1 over x2-x1

What does C represent in the rule of a constant function?

a number.

f(x) = 16x^2 + 32x, find out the: a) Concavity b) y-intercept c) Vertex d) Maximum or a minimum

a) "a" = positive (concave up) b) There is no "c"! (c is just equal to 0 hence) Therefore, the y-intercept is: (0, 0) c) Use the formula! Remember c = 0: V --> (-b/2a , 4ac - b^2/4a) V --> (-32/2(16), 4(16)(0) -(32)^2/4(16) V = (-1 , -16) d) Concave up, it looks like it's smiling, therefore the VERTEX is a MINIMUM.

f(x) = 1/2x - 6 a) Slope? b) y-intercept? c) x-intercept?

a) 1/2 b) "b" is the y-intercept: -6. But we want a POINT: (0, -6) c) The "x" intercepts when "y" is on 0 (visual a graph) Replace f(x) by 0 0 = 0.5x - 6 6 = 0.5x x = 12 GIVE ANSWER AS A POINT: (12, 0)

If f(x) = 2x^2 + 8x + 30, find the: a) Concavity b) y-intercept c) Vertex

a) For the concavity, we look at "a" a = 2; and since "a" is positive, we can thus conclude that it is concave up (^). In a graph, it would look like a smile. b) To get the y-int., all we need to do is look at "c". c = 30 And since we want a POINT, y-intercept: (0, 30) c) For the vertex, we need actual work. Use the formula of a vertex: V --> (-b/2a , 4ac - b^2/4a) And plug everything in. V --> (-8/2(2) , 4(2)(30) - (8)^2/4(2)) V = (-2, 22) (Represent it as a POINT). *Signs CAN change if negative (two negatives make a positive)

What is the rule of a constant function?

f(x) = C

What is the rule of the quadratic in the standard form?

f(x) = a(x-h)^2 + k

What are the x-intercepts of f(x) = 5(x-1)^2 + 25?

f(x) = a(x-h)^2 + k Look at "a" and "k" and see if they have opposite signs. They don't, so no x-intercepts! BUT, if you wanted to do it the long way, you can see that it doesn't work: 0 = 5(x-1)^2 + 25 -25/5 = 5(x-1)^2 -5 = (x-1) You can't square root the 5!! So you see? The x-intercepts D.N.E. (does not exist)

In the standard form, what point is the vertex?

f(x) = a(x-h)^2 + k The vertex is the point (h , k)

If f(x) = 16(x-180)^2 + 1020, what is the equation of the axis of symmetry?

f(x) = a(x-h)^2 + k Use the formula for the axis of symmetry: x = h x = 180 (Remember "h" is always the opposite of what we see)

If f(x) = 2(x-1)^2 +4, what is the vertex of f(x)?

f(x) = a(x-h)^2 + k We know that the vertex is the point (h,k). We also know that "h" is always the opposite of what we see. So the point of the vertex is (1,4)

What is the rule for the direct function?

f(x) = ax

What is the rule for a partial function?

f(x) = ax + b

What is the rule for the general form of the quadratic?

f(x) = ax^2 + bx + c

What is a vertex?

the minimum or maximum point of parabola.

What is the equation of the axis of symmetry in the general form?

x = -b/2a (this will give you the vertical line)

What formula do we use to find the x-intercepts for the standard form?

x = h +/- (square root): -k/a *Remember that "h" is ALWAYS the opposite of what we see!

What's a trick to know what to do to solve for the x-intercepts and the y-intercepts?

x-intercepts: You would replace f(x) by 0, think "isolating" the x. THIS DOESN'T WORK FOR THE GENERAL FORM!! y-intercepts: You would replace "x" by 0! Think isolating the "y"