math grade 9 study guide

7/29/2019 Math Grade 9 Study Guide

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Grade 9 Academic Math Study Guide

Dan Petrenko, 2012


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4.1- Direct Variation

A direct variation is when the independent variable is multiplied/divided bya specific number; the dependent variable is ultimately multiplied/divided by thesame number.

In direct variation, the ratio of the corresponding values of the two variablesdoes not change. This ratio is called constant of variation. Ifyvaries directly withx,the equation is in the form of:y=mxwhere mis the constant of variation.

On a graph, we know two variables have a direct variation when:1) The points form a straight line

2) The line goes through the originEx.) This Scatter Plot shows the pay in dollars Tom receives for each hour heworks at Wal-Mart.

Is this a direct variation? Why?

Identify what each variable in the equation (y=mx) stands for relating it tothe above scenario:

Identify the constant of variation.

Using the direct variation equation, Identify Toms pay for the 18th hour thathe works. (Remember to write a therefore statement.)

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Pay ($)

Pay ($)


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4.2- Partial Variation

A partial variation is when the independent variable is multiplied/divided bya specific number; the dependent variable is NOT multiplied/divided by the samenumber.

A partial variation is when the two variables relate using a constant ofvariation (like in direct variation) and an initial value.

Ifyvaries withxpartially, the equation is in the form of:y=mx+b where m is

the constant of variation and b is the initial value (most often called they-intercept.)

On a graph, we know two variables have a partial variation when:1) The points form a straight line

2) The line DOES NOT go through the originDirect Variation Partial Variation

Definition One variable is the constantmultiple of the other one. (m is aconstant multiple of x.)

The dependent variable is thesum of a constant number (b)and a constant multiple (m) ofthe independent variable.

Table of Values (0,0) is an ordered pair. (0,b) is an ordered pair.Graphs Straight line

Line goes through the origin(0,0).

Straight lineLine wont go through the

originy-intercept is (0,b)

Equations y=mxm constant of variation

y=mx+bm constant of variationb initial value/ y-intercept

Ex.) Billy Bob sells magazines at a kiosk. He gets paid 6 dollars for the hour that heworks and 3 more dollars for each magazine that he sells in the hour. This graphrepresents his sales for one hour.

1. Identify the initial value and the constant of variation for the above graph.

2. Write the equation for the line in the proper partial variation standard(y=mx+b.)

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Y-Value

Y-Value


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4.3- Slope

The steepness of a line segment is measured by its slope.The slope is the ratio of the rise to the run and is often represented by the letter m.The rise is the vertical distance between two points, while the run is the horizontaldistance between two points.

Rise (Change in y)m= ------

Run (Change in x)

A line segment rising from left to right has a positive slope. A line segment falling from left to right has a negative slope. A horizontal line has a zero slope. A vertical line has an undefined slope.

Solving slope to get the coordinates of the line:

Graphical Method-1. Plot the first point.2. Use the slope (rise over run)- move up/down on the vertical axis depending

on the rise and move left/right on the horizontal axis depending on the run.3. Identify the coordinates of the second point.4. Connect the two points to create a line.

Numerical Method:1. Add the run to the x-coordinate2. Add the rise to the y-coordinate3. Identify the second point.

Ex.) Find the slopes of the lines below. (Remember- your final answer should looklike: m=rise/run.)


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4.4- Rate of Change

Definition- The change of 1 quantity relative to anotherEx) - cost per pound

- Km per hour- Cost per gram- Population per year- cm per second- Pay per hour- Cost per liter

Ex.) Solve the following question using a rate of change as your units.

The average adult breath is about 40 liters of air every 5 minutes. What is the rate ofchange of volume of air?

The following equation represents the total costC, in dollars, of hiring a gardener forthours: C= 25t+15

a) Identify the constant of variation in the above equation.

b) Interpret the constant of variation as a rate of change.

Use the graph below to write your own question involving a constant of variationand rate of change.


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4.5- Equation of a line in Slope y-Intercept Form

The equation of the line (EOTL) in slope y-intercept form is written as: y=mx+bwhere:y dependent variablex independent variablem slopeb y-intercept

Slope & y-Intercept in an Equation-Find the slope (m) bye: looking for the number multiplying the independentvariable (coefficient.)

Find the y-intercept (b) by: looking for the number that is added or subtracted(constant.)

Slope & y-Intercept on a Graph-Find slope (m) by:

Finding two points Drawing a right angle m= rise / run

Find y-Intercept (b) by:

Finding the y-value where the line intersects the y-axis.Ex.) Identify the slopes and y-intercepts:

Equation Slope (m) y-Intercept (b)

a) y= 2x+5b) y= x +4c) y= -3x +15

d) y= 40x -12

e) Slope (m):_______________

y-Intercept (b):________________

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4.6- Equation of a line in Standard Form

The EOTL in Standard Form is written as: Ax+By+C=0 A, B, and C must me integers. A & B are both NOT 0. A is always positive.

Converting Equations in Standard Form to Slope y-Intercept Form: (Isolating y)

1. Subtract Ax and C from both sides.2. Divide every term by B (ys coefficient)3. Simplify.

Ex.) Convert these equations to Slope y-Intecept Form:

a) 3x+5y+13=0

b) 15x+2y+42=0

c) 20x+3y+12=0

4.7- Graphing Lines & x-Intercepts, y-Intercepts:

A y-intercept is the point where the line crosses the y-axis.Similarly, and x-intercept is the point where the line crosses the x-axis.

All y-Intercepts are in the form of (0, b)All x-Intercepts are in the form of (a, 0)

Diagonal Lines have an x and y Intercept. Horizontal Lines have only a y-Intercept Vertical Lines have only an x-Intercept.For x-Intercept, y is always zero.

For y-intercept, y is always zero.

By subbing in x=0 into the equation, you can solve for y, which is the y-intercept ofthe line.By subbing in y=0 into the equation, you can solve for x, which is the x-intercept ofthe line.


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Ex.) Solve.

1. y = 4x +12X=0 to find y. Y=0 to find x.

2. y = - x +8X=0 to find y. Y=0 to find x.

3. Is it possible to graph a line if you are given an x-intercept of 0 and a y-intercept of 0? Explain.

4.8- Solving Systems of Equations by Graphing:

To solve a system of equation means finding out where the lines intersect. Tosolve a system of equation by graphing, you need to graph each of the lines and thenstate the coordinate/point where they intersect/meet.

Ex.) Solve these systems of equations.


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4.9- Parallel and Perpendicular Lines:

Parallel lines are lines that are always the same distance apart. They mayhave different y-intercepts, but ALWAYS have the same slope.

Perpendicular lines are lines that intersect at a right angle. They may havedifferent y-intercepts, but the slopes are ALWAYS negative reciprocals.

The product of the slopes is equal to -1.

Ex.) State whether the following lines are parallel, perpendicular, or neither.

a) y= x-15 & y= -4x +2b) y= 32x+4 & y= 23x +4

c) y= x + 20 & y= x + 5

d) y= -2x +13 & y= - x +6


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4.10- Finding the EOTL Given a Slope and a Point:

Steps for finding the EOTL:1. Start with y=mx+b and plug in whatever is already given. (Usually the slope.)2. Plug in the given point for x and y.3. Isolate for b.4. Write the equation of the line, including the found b value.

Ex.) Determine the EOTL that has the following conditions.

a) Is parallel to y= 4x+7 and goes through (5, -2)

b) Is perpendicular to y= -3x -2 and passes through (-1,4)

c) Has the same slope as 6x +3y-21=0 and passes through (7, -11)

d) Graph the above lines on the grid below and label them a, b, & c.


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4.11- Finding the EOTL Given Two Points:

Steps for finding the EOTL:1. Find the slope using the equation (m= y2-y1/x2-x1)2. Plug the slope into the equation y=mx+b3. Plug in the coordinates from any point that is given into y=mx+b (as y and x)4. Isolate the y-Intercept (b)5. Plug in value of b into the equation

Ex.) Given two points, find the equation of the line.

a) (2,3) and (-2,6)

b) (-4,0) and (6,5)

c) (-7, 5) and (4,3)

d) Graph the above lines on the grid below and label them a, b, & c.

math grade 9 study guide

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