1. Linear Equations & Functions

2. Evaluate the following functions
If f(x) = 3x2 + 2x
f(-1)
f(-4)
f(-2)
f(2) + 5
2 f(-3)

3. Forms of a Linear Equation
Standard Form
Ax + By = C
(where A,B, and C are real numbers and A and B are not both 0)
Slope-Intercept Form
y = mx + b
( m is the slope and b is the y-intercept)
Point-Slope Form
y – y1 = m( x1 – x )

4. What are the intercepts of a linear equation?
The x-intercept is the x-coordinate of the point where the graph (in this case the line) crosses the x-axis. The y-coordinate of this point is always 0. The point is described as (x,0).
The y-intercept is the y-coordinate of the point where the graph (in this case the line) crosses the y-axis. The x-coordinate of the point is always 0. The point is described as (0,y).
* Every line will not have both intercepts. A Horizontal line will not have an x-intercept. A vertical line will not have a y-intercept.

5. How do you find the x and y intercepts?
You find the x-intercept by making y = 0 and solving the equation for x.
Find the x-intercept of 3x – 2y = 6
3x – 2(0) = 6
3x = 6
x = 2
2 is the x-intercept.
The point is (2,0).
You find the y-intercept by making x = 0 and solving the equations for y.
Find the y-intercept of 3x – 2y = 6
3(0) – 2y = 6
-2y = 6
y = -3
-3 is the y-intercept.
The point is (0,-3).

6. How can we use the intercepts to graph the line?
This Photo by Unknown Author is licensed under CC BY-SA

7. Let’s Practice!
Find the x and y intercepts of the following equations.
1. 8x + 7y = 28
x – 8y = 24
3. -5x + 6y = 60
Complete pg. 177 #3-18

8. Slope Intercept Form

9. How do you change a linear equation from standard form to slope-intercept form? What information does this form give you about the line?
Solve the equation in terms of y.
Example 1: 8x + 4y = -36 Subtract 8x from both sides.
4y = -8x – 36 Divide each term by 4 to get x by itself.
y = -2x – 9 Slope-Intercept Form
The slope is -2 and the y-intercept is -9
Example 2: 3/2 x = -4y – 12 Multiply each term by 2 to eliminate the fraction.
3x = -8y – 24 Add 24 to each side.
3x + 24 = -8y Divide each term by -8 to get y by itself.
-3/8x – 3 = y
y = -3/8x – 3 Slope-Intercept Form
The slope of the line is -3/8 and the y-intercept is -3.

10. Let’s Practice!
Write the following equations in slope-intercept form. Then identify the slope and y-intercept.
1. 9y = 3x + 18
2. 8x + 4y = -56
3. -4x + 2y = 10
4. 3x – 2y = -12

11. Slopes & Types of Lines
Line (rising from left to right) Positive Slope y = mx + b
Line ( falling from left to right) Negative Slope y = -mx + b
Horizontal Line (0/any number) Slope is 0 y = any number
Vertical line (any number/0) Slope is undefined x = any number

12. Finding the Slope
The slope o a line is the ratio of rise to run for any two points on the line.
The rise is the difference in the y-values and the run is the difference in the x-values of two points on a line
Slope = rise Slope = difference in y
run difference in x
If (x1 , y1) and (x2 , y2) are two points on a line, the slope of the line is found by using the slope formula.
Slope Formula m = y2 - y1
x2 - x1

13. Calculating the slope using the slope formula
Example 1
Find the slope of the line containing (3,4) and (2,1)
M = 1 – 4 = -3 = 3
M = 3
Example 2
Find the slope of the line containing (-2,3) and (-4, 6)
M = 6 – 3 = 3
-4 –(-2) -2
M = -3/2

14. Let’s Practice! Find the slope of each line 1. (2, 3) and (-2, -1)
Complete pg. 184 #1 -12 in text.

15. Writing the Equation in Slope-intercept Form given the slope and a point on the line
Write the equation of the line with a slope of 3 that goes through the point (2,5).
First, substitute the point (x,y) and the slope in the slope-intercept form and solve for b.
x = 2 y = 5 m = 3
Slope-Intercept Form y = mx + b
5 = 3(2)+ b
5 = 6 + b
-1 = b
Slope-Intercept Form
y = 3x -1

16. Writing the Equation of a line in Slope-intercept Form Given Two Points
Write the equation of the line (slope-intercept form) going through (0,5) and (2,13).
First, use the slope formula to find using the given points
m = y2 – y1 m = 13 – 5 8 m = 4
x2 – x
Second, use the slope and one of the points in the slope-intercept form to find b (y-intercept). m = 4 point (0,5) x = 0 y = 5
y = mx + b
5 = 4(0) = b
5 = b
Lastly, use the slope and the y-intercept to write the equation of the line.
y = 4x + 5

17. Let’s Practice! Write the equation of the line in slope-intercept form
1. m = -1/2 going through ( -4,6)
2. going through (1,6) and (2,3)
3. m = - ¼ b = - 8
4. -3x + 7y = 21

18. Point –Slope Form of a Linear Equation

19. Writing an Equation in Point-Slope form when given the slope and a point on the line
The point slope form of a linear equation y – y1 = m( x – x1 )
1. Write the equation in point-slope form if the slope is 3 and the point on the line is (-5, 2).
First, identify m, x1 and y1 and then substitute these values in the equation.
m = 3 x1 = -5 y1 = 2
Point-slope Form: y – 2 = 3(x –(-5))
y – 2 = 3(x + 5)

20. Writing an Equation in Point-Slope Form when given two points
1. Write an equation of the line in point-slope form if (2,1) and (3,4) are on the line.
First, substitute the points in the slope formula and find the slope. m = y2 – y1
x2 – x1
m = 4 – 1 = 3
3 – 2 1
Next, use the slope and any one of the points and substitute it in the point-slope form.
Point-Slope Form y – 1 = 3( x – 2)

21. Let’s Practice! Write an equation in point-slope form.
1. The slope is 2 and the point on the line is (-2,2)
2. The slope is 5 and the point on the line is (-2,4).
Write an equation in point-slope form.
1. The points on the line are (4,-7) and ( 2, -3).
2. The points on the line are (1,5) and (-10, -6).

22. Changing Point-Slope Form into Standard Form
Write the equation in standard form y – 5 = -2( x + 4 )
First, distribute and change to slope-intercept form
y – 5 = -2x – 8 Add 5 to both sides of the equation.
y = -2x -3 Slope-Intercept Form
Next, get x and y on the same side of the equation.
y = -2x – 3 Add 2x to both sides of the equation.
2x + y = -3 Standard Form

23. Let’s Practice! Write the following equations in standard form.
1. y – 3 = 4( x – 3 )
2. 6 – y = -5x + 6
3. y – 10 = 3( x – 5)

24. Slopes of Parallel & Perpendicular Lines
When two lines are parallel, their slopes are equal. For example, if the equation of a line is y = -3x + 2, any line parallel to this line will also have a slope of -3. If the slope of a line is y = -2/5 x – 7, any line parallel to this line will also have a slope of -2/5.
When two lines are perpendicular, their slopes are negative reciprocals. For example, if the equation of a line is y = 3x + 2, the slope of a line perpendicular to this line will be – 1/3. If the equation of a line is y = 1/2x – 8, the slope of the line perpendicular to this line will be -2.

25. Writing the Equations of Parallel lines
Write the equation of a line that is parallel to y = 5x + 2 and goes through the point (-1, 3 ).
A line parallel to the line above will also have a slope of 5. Therefore, we will substitute 5 as the slope (m) and the point given in the slope-intercept form to find b and then write the equation of the line.
y = mx + b
3 = 5(-1) + b
3 = -5 + b
8 = b
Slope- Intercept Form y = 5x + 8

26. Writing the Equations of Perpendicular LInes
Example 1
Write the equation of a line that is perpendicular to y = - 1/4x + 3 and goes through the point ( -2, 6).
A line perpendicular to the line above will have a slope of 4 (which is the negative reciprocal of -1/4). We need to use this slope and the point given in the slope-intercept form to find b.
Y = mx + b
6 = 4(-2) + b
6 = -8 + b
14 = b
Slope-Intercept Form y = 4x + 14
Example 2
Write the equation of a line that is perpendicular to y = 3x – 10 and goes through the point ( -6, 4 ).
A line perpendicular to the line above will have a slope of -1/3 (which is the negative reciprocal of 3). We need to use this slope and the point given in the slope-intercept form to find b
Y = mx + b
4 = -1/3 (-6) + b
4 = 2 + b
2 = b
Slope-intercept Form y = -1/3x + 2

27. Let’s Practice!
1. Write the equation of the line that is parallel to y = 2x + 3 that goes through ( -1, 4).
2. Write the equation of the line that is parallel to y = -3/4 x – 2 that goes through ( -8, -2 ).
3. Write the equation of the line that is perpendicular to y = -3/ that goes through ( 3, 5 ).
4. Write the equation of the line that is perpendicular to y = 4x -6 that goes through (4, -4).