Identify Linear Functions & Their Graphs Honors Math – Grade 8

Identify Linear Functions & Their GraphsHonors Math – Grade 8

Get Ready for the LessonIt is recommended that no more than 30% of a

person’s daily caloric intake come from fat. Since each gram of fat contains 9 Calories, the most grams of fat f that you should have each day is given by

30Cf

C is the total number of Calories that you consume. The graph of this equation is shown to the right.

This type of equation is known as a linear equation.

A linear equation is the equation of a line.

KEY CONCEPT Standard Form of a Linear EquationThe standard form of a linear equation is

where A > 0, A and B are not both zero and A, B, and C are integers with a greatest common factor of 1.

CByAx

3295 yx

6x

42 yx

0y

Determine whether each equation is a linear equation. If so, write the equation in standard form.

xy 25 Rewrite the equation so that it is in the form:

Ax + By = C

Both variables need to be on the same side of the equation.

Add 2x to both sides of the equation.

xx 22

52 yx

This equation is in standard form where A = 2, B = 1 and C = 5

This equation is a linear equation.


652 yxyRewrite the equation so that it is in the form:

Ax + By = C

Since the term 2xy has two variables, the equation cannot be written in the form Ax + By = C.

This equation is not a linear equation.

235 zyx Rewrite the equation so that it is in the form:

Ax + By = C

Since the equation has a third variable—z, it cannot be written in the form Ax + By = C.



32 xyRewrite the equation so that it is in the form:

Ax + By = C

Since the equation contains an x2, it cannot be written in the form Ax + By = C.


131

yEliminate the denominator by multiplying both sides of the equation by 3.

3 3

3yThis equation is in standard form where A = 0, B = 1 and C = -3

This equation is a linear equation.

The x-coordinate of the point at which the graph of an equation crosses the x-axis is an x-intercept.

The y-coordinate of the point at which the graph of an equation crosses the y-axis is a y-intercept.

The coordinates of an x-intercept are (x, 0). An x-intercept occurs when y = 0.

The coordinates of a y-intercept are (0, y). An y-intercept occurs when x = 0.

Values for x for which f(x) = 0 (AKA y = 0) are called zeros of the function. The zero of a linear function is its x-intercept.

Graph each equation

321

xyThe graph of an equation represents all of its solutions. So, every ordered pair that satisfies the equation represents a point on the line. An order pair that does not satisfy the equation represents a point not on the line.

To graph an equation, make a function table.

x y Ordered Pair

321

x

Choose values for the domain (x).

-2

0

2

4

Choose at least three numbers.

Evaluate for each value and write the ordered pair.

3)2(21

3)0(21

3)2(21

3)4(21

-4

-3

-2

-1

(-2,-4)

(0,-3)

(2,-2)

(4,-1)

Plot the points and connect them with a line.

Graph each equation

2yThis equation states that y is always -2.

To graph an equation, make a function table.

x y Ordered Pair

Choose values for the domain (x). Remember that no matter what you choose, y is always -2.

-1

0

1

2

Choose at least three numbers.

An equation in the form y = C is the equation of a horizontal line.

-2

-2

-2

-2

(-1,-2)

(0,-2)

(1,-2)

(2,-2)

Plot the points and connect them with a line.

Graph each equation

923 yx An equation can also be graphed using the x- and y-intercepts of the equation.

To find the x-intercept let y = 0.

923 yx9)0(23 x93 x3x

)0,3(

To find the y-intercept let x = 0.

923 yx92)0(3 y92 y5.4y

)5.4,0(

Plot the ordered pairs and connect them with a line.

Graph each equation

4 xy An equation can also be graphed using the x- and y-intercepts of the equation.

To find the x-intercept let y = 0.

4 xy40 x

4x

)0,4(

To find the y-intercept let x = 0.

4 xy4)0(1 y

4y

)4,0(

Plot the ordered pairs and connect them with a line.

Rate of change is a ratio that describes, on average, how much one quantity changes with respect to a change in another quantity. If x is the independent variable and y is the dependent variable, then

The table below shows the distance a person has walked for different amounts of time.

What would the rate of change be?

14 This means the person

walked four feet per second.

Use the table to find the rate of change. Explain the meaning of the rate of change.

The rate of change means that it costs $39 per game.

xinchangeyinchange

2478156

278

139

Find the rate of change.

The slope of a line is the ratio of the change in the y-coordinates (rise) to the change in the x-coordinates (run) as you move in the positive direction.

The graph shows a line that passes through (1, 3) and (4, 5).

32

slope Any two points on the line can be used to determine the slope.

KEY CONCEPT SlopeThe slope of a line is the ratio of the rise to the run.

12

12

xxyym

The slope m of a nonvertical line through any two points, (x1, y1) and (x2, y2), can be found as follows:

Find the slope of the line that passes through the given points.

(-1, 2) and (3, 4)Define the variables.

Use the slope formula. 12

12

xxyym

1324

Substitute and simplify.

42

21

(3, 6) and (4, 8)Define the variables.


12

xxyym

3468


12

2

The slope of a line may be positive!


(-1, -2) & (-4, 1)Define the variables.


12

xxyym

1421


33 1

(-2, 2) & (-6, 4)Define the variables.


12

xxyym

2624


42 2

1

The slope of a line may be negative!


(6, 7) and (-2, 7)Define the variables.


12

xxyym

6277


80 0

(1, 2) and (-1, 2)Define the variables.


12

xxyym

1122


20

0

The slope of a line may be zero!


(1, 2) and (1, 4)Define the variables.


12

xxyym

1124


02

(3, -2) and (3, 2)Define the variables.


12

xxyym

3322


04

Since division by zero is undefined, the slope is undefined.

Since division by zero is undefined, the slope is undefined.

Given the slope of a line and one point on the line, you can find other points on the line using the slope formula and your expertise in solving equations!

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Identify Linear Functions & Their Graphs Honors Math – Grade 8