Math 9th Grade
Inequalities Equations
-X+3>2 Greater than -7x+1≤36 Less than or Equal Too Step 1 Eliminate fractions by multiplying all terms by the least common denominator of all fractions. Step 2 Simplify by combining like terms on each side of the inequality. Step 3 Add or subtract quantities to obtain the unknown on one side and the numbers on the other.
How to solve an equation (With variables)
1. Distribute on both sides 2. Combine like terms on each side 3. Move the variable across the equals sign so they are together (if there are variables on both sides) 4. add & subtract the constant to move it away from the variable (change the sign) 5. Divide or multiply the coefficient to move it away from the variable (don't change signs)
Scatter Plots and Correlation
A cluster of dots to represent two variables. Correlation- relationship to each other. A scatterplot displays the strength, direction, and form of the relationship between two quantitative variables. A correlation coefficient measures the strength of that relationship. Calculating a Pearson correlation coefficient requires the assumption that the relationship between the two variables is linear.
What is Factored form and Expanded form
A factored form is a parenthesized algebraic expression. In effect, a factored form is a product of sums of products or a sum of products of sums. Any logic function can be represented by a factored form, and any factored form is a representation of some logic function. Expanding from factored form uses the distributive property of real numbers which states for real numbers x and y, c (x + y) = cx + cy. Example. Expand and simplify x (x2 + 5). We apply the distributive property to the equation to expand and then we simplify the resulting equation.
Scatter Plot Graph
A graph of plotted points that show the relationship between two sets of data that do not depend on each other where the line is not connected. (Ex. height versus weight)
How to know if something is quadratic
A quadratic equation is an equation of the second degree, meaning it contains at least one term that is squared. For example 5x^2 that is quadratic.
Quadratic Functions
A quadratic function is one of the form f(x) = ax2 + bx + c, where a, b, and c are numbers with a not equal to zero. The graph of a quadratic function is a curve called a parabola. Parabolas may open upward or downward and vary in "width" or "steepness", but they all have the same basic "U" shape.
What is Range and Domain
Because the domain refers to the set of possible input values, the domain of a graph consists of all the input values shown on the x-axis. The range is the set of possible output values, which are shown on the y-axis. Consider a function q defined by g(x) = x². Explain why negative values are not included in the range of q. When any number is squared, the result is positive or zero. Because g(x) = x² will produce only positive outputs or zero, the range is all values of q(x) that are 0 or greater.
how to write a percent
For example 24/25 would be 96% One step, you divide 24 by 25. You would get a decimal of 0.96 Another example is 15/35 You divide 15 by 35 you would get 0.42857142857. But really you only need the first 2 decimal places, in this case it is the .42. So that is 42%. But if you want the special numbers after .42, which is 85. You would round up in this case making them 86. You would simply put a decimal place between them and the 42. Final answer is 42.86%
How do you find the rate of change from a graph
For example radioactive substance over a 10 year period You are going to find the starting point then you are going to find the second point. Next you are going to put the second point over the starting point(Does not matter whether it is decreasing or increasing) That will create a fraction This will determine the average rate of change, Simplify if you can. Boom thats your rate of change. The year 0 was 30 grams, then the next year was 25. 25/30= 5/6. Formula would be 30(5/6)^t On average it changed -2.4 per year for the 10 year period. For every year change is 5/6. (t=year)
Parabola Quadratic Inequality Graphing
Graph the quadratic inequality. y≤x2−x−12 The related equation is: y=x2−x−12y=x2−x−12 First, we notice that A, the coefficient of the x^2 term, is equal to 11. Since A is positive, the parabola points upward. The right side can be factored as: y=(x+3)(x−4)y=(x+3)(x−4) So the parabola has X -intercepts at −3 and 4. The vertex must lie midway between these, so the X -coordinate of the vertex is 0.5 Plugging in this X -value, we get: y=(0.5+3)(0.5−4) y=(3.5)(−3.5) y=−12.25 So, the vertex is at (0.5,−12.25). Should you shade the region inside or outside the parabola? The best way to tell is to plug in a sample point. (0,0) is usually easiest: 0≤?0^2−0−12 0≤−12 So, shade the region which does NOT include the point (0,0).
How to write an Equation (With variables)
Identify the unknown and represent it with a variable. Set up an equation or inequality using that variable. Solve the equation or inequality to find an answer to the problem.
What is Quadratic in Mathematics
In mathematics, the term quadratic describes something that pertains to squares, to the operation of squaring, to terms of the second degree or equations or formulas that involve such terms. In other words, the quadratic formula is simply just ax^2+bx+c = 0 in terms of x.
Slopes and Intercepts
In the equation of a straight line (when the equation is written as "y = mx + b"), the slope is the number "m" that is multiplied on the x, and "b" is the y-intercept (that is, the point where the line crosses the vertical y-axis). Two forms Standard form: Ax+By=C. Slope-Intercept Form: y=mx+b
Line Of Best Fit
Line of best fit refers to a line through a scatter plot of data points that best expresses the relationship between those points. A straight line will result from a simple linear regression analysis of two or more independent variables.
How to multiply decimals
Multiply normally, ignoring the decimal points. Then put the decimal point in the answer - it will have as many decimal places as the two original numbers combined. In other words, just count up how many numbers are after the decimal point in both numbers you are multiplying, then the answer should have that many numbers after its decimal point.
How do outliers affect the mean median and standard deviation?
Outliers affect the mean value of the data but have little effect on the median or mode of a given set of data. A single outlier can raise the standard deviation and in turn, distort the picture of the spread. For data with approximately the same mean, the greater the spread, the greater the standard deviation. If all values of a data set are the same, the standard deviation is zero (because each value is equal to the mean).
Inequality Graphing
Rearrange the equation so "y" is on the left and everything else on the right. Plot the "y=" line (make it a solid line for y≤ or y≥, and a dashed line for y< or y>) The shade above the line for a "greater than" (y> or y≥) or below the line for a "less than" (y< or y≤).
difference between standard and expanded form
Standard form is a way of writing down very large or very small numbers easily. The expanded form is Writing a number to show the value of each digit. It is shown as a sum of each digit multiplied by its matching place value
How to divide decimals
Step 1: Estimate the answer by rounding. You'll use this estimate to check your answer later. Step 2: If the divisor is not a whole number, then move the decimal place n places to the right to make it a whole number. Then move the decimal place in the dividend the same number of places to the right (adding some extra zeros if necessary.) Step 3: Divide as usual. If the divisor doesn't go in evenly, add zeros to the right of the dividend and keep dividing until you get a 0 remainder, or until a repeating pattern shows up. Step 4: Put the decimal point in the quotient directly above where the decimal point now is in the dividend. Step 5: Check your answer against your estimate to see if it's reasonable.
Multiplying Fractions
Step 1: Multiply the numerators from each fraction by each other (the numbers on top). The result is the numerator of the answer. Step 2: Multiply the denominators of each fraction by each other (the numbers on the bottom). Step 3: Simplify or reduce the answer.
Dividing Fractions
The first step to dividing fractions is to find the reciprocal (reverse the numerator and denominator) of the second fraction. Next, multiply the two numerators. Then, multiply the two denominators. Finally, simplify the fractions if needed.
how do you solve for x in an equation and inequality
The first step to solving for x is going to be getting x alone on one side of the equation and everything else on the other side. Remember the algebraic golden rule: What you do to one side of the equation, you must do to the other side. That's how the equation stays equal!
What is a parabola
The graph f(x) is called a Parabola. Examples: f(x) = 2x^2 - x + 1 (Opens Upward because positive. a>0.) g(x) = -x^2 + 3x -5 (Opens Downward because negitive. a<0.) The highest (Absolute Max)/lowest (Absolute Min) point is called the Vertex.
How do you add or subtract percentages to real numbers
The marked price of a shirt is 7200 a discount of 15% is on sales what is the sale price 1) 15/100 × 7200 =1080 ( the amount of discount based on the marked price ) 2) 7200-1080=6120
Standard deviation, Mean, Mode, Median
The mean, median, and mode are all estimates of where the "middle" of a set of data is. The standard deviation is the average distance between the actual data and the mean.
Finding the Area and Perimeter
The perimeter P of a rectangle is given by the formula, P=2l+2w, where l is the length and w is the width of the rectangle. The area A of a rectangle is given by the formula, A=lw, where l is the length and w is the width.
Inequalities System of Equations
The solution of the system of inequalities in the intersection region of all the solutions in the system. For example graph the inequality y≤x−2. The related equation is y=x−2. The solution of the system of inequalities in the intersection region of the solutions of the two inequalities. (This is just an example photo)
System of Equations
There are three ways to solve systems of linear equations: substitution, elimination, and graphing. Elimination: Step 1: Multiply each equation by a suitable number so that the two equations have the same leading coefficient. Step 2: Subtract the second equation from the first. Step 3: Solve this new equation for y. Step 4: Substitute y = 2 into either Equation 1 or Equation 2 above and solve for x. Substitution: Step 1: Solve one equation for one of the variables. Step 2: Substitute (plug-in) this expression into the other equation and solve. Step 3: Resubstitute the value into the original equation to find the corresponding variable.
how to convert slope intercept to standard form
To convert from slope intercept form y = mx + b to standard form Ax + By + C = 0, let m = A/B, collect all terms on the left side of the equation and multiply by the denominator B to get rid of the fraction
Basic Rules of Graphing Quadratic Inequality (Parabola)
To graph, a quadratic inequality, start by graphing the parabola. Then fill in the region either above or below it, depending on the inequality. If the inequality symbol is ≤ or ≥ , then the region includes the parabola, so it should be graphed with a solid line. Otherwise, if the inequality symbol is < or >, the parabola should be drawn with a dotted line to indicate that the region does not include its boundary.
How to identify Y-intercepts without graphing(Expression form)
To solve this problem, we need to start by expanding the expression, meaning we multiply (x-7) by (x+3). The most common method for this is FOIL (first, inside, outside, last), but you may be familiar with another technique. x*x + x*3 + -7*x + -7*3 x^2 + 3x -7x -21 x^2 - 4x - 21. Then you plug in 0 as X that will get you the y intercept of -21
Vertex
Vertex the highest or lowest point in a graph
Linear vs Non-Linear
When graphed, a linear equation will have a straight line that has a constant slope. In contrast to this, a nonlinear equation will have a graph that does not have a straight line and, depending on the function, can have many different appearances including a U-shape or an S-shape.
Function Notation
an equation in the form of 'f(x)=' to show the output value of a function, f, for an input value x. You need to remember that function notation simply uses f(x) to denote y. Therefore if we can write an equation in slope-intercept form, y=mx+b then we can change that to function notation. Remember in slope-intercept, y=mx+b , m represents the slope and b represents the y-intercept.
Pythagorean theorem
a²+b²=c² Is the formula The Pythagorean Theorem is a fundamental relation in Euclidean geometry among the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse is equal to the sum of the areas of the squares on the other two sides.
Distance between two points
d = √(x₁-x₂)² + (y₁-y₂)²
How to write a function
f(x)=mx+b f(x) is whatever the amount you trying to find M is the beginning amount that will be decreasing or increasing X is the rate at which the function is changing, negative or positive. B if there are any additional amount to be considered (optional) Things such as consumer rates or gravity or even height advantages For example h(t) = 5 + 20t −16t^2 h(t) is the time over seconds what height would the ball be? H=height T=time +5 is the additional height off the ground 20t is the verticle rate/speed at which the ball was thrown over time. The T will determine what height the ball would be over T seconds. For example, 20(2) After 2 seconds since the ball was thrown it was 40 feet in the air(not including gravity) -16t^2 is the gravity acting on the ball. The 40 feet is without gravity with gravity if you do the math 5+20(2)-16t^2 the height will decrease from 40. (-24, the ball already hit the ground) So when h(2) the ball is thrown after 2 seconds the ball would have already hit the ground.
Exponential Rate of Change
growth whose rate becomes ever more rapid in proportion to the growing total number or size. For example: y=3^x+26
Function of Revenue scatter plot with support from table
https://docs.google.com/document/d/1br4mocqp_90iSXfWI0yWxbMAIHUmUUZRwzZpWayT9W8/edit
Quadratic graph looks like
https://docs.google.com/document/d/1br4mocqp_90iSXfWI0yWxbMAIHUmUUZRwzZpWayT9W8/edit
How to solve for X exact form (Quadratic)
https://www.google.com/search?q=How+to+solve+for+X+exact+form+%28Quadratic%29&safe=strict&sxsrf=ALeKk00EuYbepb9cB79shQgWmeSznHGLbQ%3A1620236061249&source=hp&ei=HdeSYKLNC42t5NoP8Ym-uAI&iflsig=AINFCbYAAAAAYJLlLbJqyjevXajTBEr03vMcJeXqU1zO&oq=How+to+solve+for+X+exact+form+%28Quadratic%29&gs_lcp=Cgdnd3Mtd2l6EAMyCAghEBYQHRAeUMEFWMEFYOAIaABwAHgAgAFZiAFZkgEBMZgBAKABAqABAaoBB2d3cy13aXo&sclient=gws-wiz&ved=0ahUKEwjiz4TXibPwAhWNFlkFHfGEDycQ4dUDCAk&uact=5#kpvalbx=_H9eSYN7uGout5NoP57qFmAg19
How to convert standard form to factored form
https://www.google.com/search?q=converting+to+factored+form&safe=strict&sxsrf=ALeKk02vq2ZyqTLo3IRKNJa_uT-_Uo2bsw%3A1620085740905&ei=7IuQYKLYNpKv5NoP7LO--AU&oq=converting+to+factored+form&gs_lcp=Cgdnd3Mtd2l6EAMyAggAMgIIADIGCAAQFhAeMgYIABAWEB4yBggAEBYQHjIGCAAQFhAeMgYIABAWEB4yBggAEBYQHjIGCAAQFhAeMgYIABAWEB46BwgAEEcQsAM6BwgAELADEEM6BAgjECc6BAgAEEM6BQgAEJECOgUIABCxAzoICAAQsQMQgwE6BwgAELEDEEM6CwguELEDEMcBEKMCOgcIABCHAhAUOggIABAWEAoQHjoFCAAQhgNQuuwOWLacD2CSoA9oA3ACeACAAekBiAH_EZIBBjI1LjIuMZgBAKABAaoBB2d3cy13aXrIAQnAAQE&sclient=gws-wiz&ved=0ahUKEwii_t3Y2a7wAhWSF1kFHeyZD18Q4dUDCA8&uact=5#kpvalbx=_54yQYNvZKozj5NoPpaqOkAs17
How to find the vertex of a parabola in standard form
https://www.varsitytutors.com/hotmath/hotmath_help/topics/vertex-of-a-parabola#:~:text=In%20this%20equation%2C%20the%20vertex,x%20%2Dcoordinate%20of%20the%20vertex. The vertex is ALWAYS between the two X-intercepts
Negative multiplying and dividing recap
A negative number times a positive number equals a negative number. When you multiply a negative number to a positive number, your answer is a negative number. It doesn't matter which order the positive and negative numbers are in that you are multiplying, the answer is always a negative number. When you divide a negative number by a positive number, your answer is a negative number. As with multiplication, it doesn't matter which order the positive and negative numbers are in, the answer is always a negative number. For example: -8 /2 = -4.
Exponents recap
A positive exponent tells us how many times to multiply a base number, and a negative exponent tells us how many times to divide a base number For example, if the is a fraction (1/4)^-2. Since we are dividing a fraction in this case the fraction would get a bigger number which is 16.
How do you find the average rate of change in a graph
A rate of change relates a change in an output quantity to a change in an input quantity. The average rate of change is determined using only the beginning and ending data. See (Figure). Identifying points that mark the interval on a graph can be used to find the average rate of change.
Parallel Line and Perpendicular
Perpendicular lines have slopes that are the opposite of the reciprocal of each other. In this case, the slope of the first line is -2. The reciprocal of -2 is -1/2, so the opposite of the reciprocal is therefore 1/2. In geometry, parallel lines are lines in a plane which do not meet; that is, two straight lines in a plane that do not intersect at any point are said to be parallel.
How to solve for X in exact form
Picture and steps. Red folder in desk
How to identify X-intercepts(Zeros) without graphing(Expression form)
The x-intercepts are 7 and -3. I rewrite the equation as (x-7)(x+3)=0. Because in this case, we want Y to equal 0, F whatever Y equals, that is why we want F=0. x-7=0 and also x+3. So if we want -7 to be 0 we must add 7. That is one X intercept, the other is 3. To make 3=0 we must subtract 3(-3) which makes it 0.
How to convert standard form to vertex form
complete the square 1, move c to the left side of the equation 2. divide the entire equation by A (to make the coefficient of x^2 be 1). 3. Take half of the coefficient of x, square it, and then add it to both sides of the equation. 4. convert the right side of the equation to a perfect binomial squared, (x + m)^2. 5. Isolate y to put the equation in proper vertex form.
residual plot
graphical representation of the residuals that can be used to determine whether the assumptions made about the regression model appear to be valid To find a residual you must take the predicted value and subtract it from the measured value. (For each point) When you plot it you keep the point on the X-axis the same, the y-axis is the only point that will be changed. for example, if you have a predicted point of (20,13) but the actual point is (20,14). You have a change of +1. The x-axis will not affect the residual. We are only focusing on the change in y. So you would plot the point (20,1) as the residual. Ideally, residual values should be equally and randomly spaced around the horizontal axis https://www.khanacademy.org/math/apstatistics/bivariate-data-ap/assessing-fit-least-squares regression/v/residual-plots
How to solve Multi-Step Equations(With Variables)
7x+3=2x+13 Clearly, our first step is to decide where to keep or isolate the unknown variable xx. Since 7x is "larger" than 2x, then we might as well keep it to the left side. This means we will have to get rid of the 2x2x on the right side. To do that, we need to subtract both sides of the equation by 2x because the opposite of +2x is −2x. After doing so, it is nice to see just the variable x on the left side. This implies that we have to move all the constants to the right side by eliminating +3+3 on the left side. The opposite of +3+3 is -3−3, therefore, we will subtract both sides by 33. The last step is to isolate the variable xx by itself on the left side of the equation. Since +5+5 is multiplying xx, then its opposite operation is to divide by +5+5. So, we are going to divide both sides by 55, and then we are done! Link to useful info for all multi-step equations: https://www.chilimath.com/lessons/intermediate-algebra/solving-multi-step-equations/
Subtracting and Adding Fractions
Check to see if the fractions have the same denominator. If they don't have the same denominator, then convert them to equivalent fractions with the same denominator. Once they have the same denominator, add or subtract the numbers in the numerator. Write your answer with the new numerator over the denominator.
Long-Division
Divide the tens column dividend by the divisor. Multiply the divisor by the quotient in the tens place column. Subtract the product from the divisor. Bring down the dividend in the ones column and repeat.
Converting expanded form to standard form (Diagram style)
Draw a box 4x4. Write both equations on each side. Link to photos https://docs.google.com/document/d/1dZXOjCrB8_dBXSIFHwOOvR9GoGaieBqMG950qBoB9Ts/edit
how to write quadratic expression into standard form
FOIL METHOD Example is (x+6)(x-2) To write the expression (x+6)(x−2) in standard form you can FOIL (First Outer Inner Last). x×x=x^2 x×(−2)=−2x 6×x=6x 6×(−2)=−12 After foiling completely you get: x^2−2x+6x−12 Next, you combine like terms. −2x+6x=4x There are no other like terms. Therefore, your final answer is x^2+4x−12
Quadratic Equations (To be Honest This is something I don't think we learned yet)
First, we bring the equation to the form ax²+bx+c=0, where a, b, and c are coefficients. Then, we plug these coefficients in the formula: (-b±√(b²-4ac))/(2a).
Zeros
Graphically, the point(s) at which a graph intersects the x-axis. Algebraically, the value(s) for which f(x)=0
Finding the Domain and Range of a Function, Another Example
f(x)=x, 4≤x<8 The points in this case since the Y-axis point is X it is unknown, so we would put the point (4,4) Closed dot, to the point (8,8) Open dot. Following the same rules of what an open dot or a closed dot would be.
Finding the Domain and Range of a Function
for example f(x)=2, -5≤x<-2 Take a plot for example, 2 in the function is the Y-axis and the -5 and -2 is the range. You can observe whether or not they have a closed or circle dot. So from this equation the answer would be a straight line from point (-5,2) Closed dot to (-2,2) Open dot ≤=Closed dot <=Open Dot Reading from left to right
FOIL Method
to multiply two binomials, find the sum of the products of the First terms, the Outer terms, the Inner terms, and the Last terms
how to find the x coordinate of the vertex of a parabola
y=ax^2+bx+c For example Plug the A and B that will get you the X-coordinate of the vertex. The vertex is always between the two x-intercepts
