Slope-Intercept Form Section 5-3 Part 1

Goals Goal Rubric To write linear equations using slope-intercept form. Level 1 – Know the goals. Level 2 – Fully understand the goals. Level 3 – Use the goals to solve simple problems. Level 4 – Use the goals to solve more advanced problems. Level 5 – Adapts and applies the goals to different and more complex problems.

Vocabulary Parent Function Linear Parent Function Linear Equation y-intercept Slope-Intercept Form

Definition Family of Functions - is a set of functions whose graphs have basic characteristics in common. Example: All linear functions form a family because all of their graphs are the same basic shape, a line. Parent Function – is the simplest or basic function within a family of functions. Linear Parent Function – is y = x or f(x) = x.

Definition Linear Equation - is an equation that models a linear function. In a linear equation the variables are raised to the 1st power only. Example: Linear equation; y = 2x + 3. Not a linear equation; y = x2 or y = 2x. The graph of a linear equation contains all the ordered pairs that are solutions of the equation. y-intercept – is the y-coordinate of the point where the line crosses the y-axis.

Slope-Intercept Form of a Linear Equation If you know the slope of a line and the y-intercept, you can write an equation that describes the line. Step 1 If a line has a slope of m and the y-intercept is b, then (0, b) is on the line. Substitute these values into the slope formula.

Slope-Intercept Form of a Linear Equation Step 2 Solve for y: Simplify the denominator. Multiply both sides by x. +b +b mx = y – b Add b to both sides. mx + b = y, or y = mx + b

Slope-Intercept Form of a Linear Equation y = mx + b slope y-intercept Any linear equation can be written in slope-intercept form by solving for y and simplifying. In this form, you can immediately see the slope and y-intercept. Also, you can quickly graph a line when the equation is written in slope-intercept form.

Finding the Slope and y-intercept of a Line Procedure for finding the slope and y-intercept of a line. Rewrite the equation of the line in slope-intercept form by solving for y. The coefficient of x, is the slope and the constant term, is the y-intercept. y = mx + b Slope (always comes in front of x!) y – intercept (where the line crosses the y-axis)

Example: Finding the Slope and the y-Intercept Find the slope and the y-intercept of the line whose equation is 2x – 3y + 6 = 0. Procedure We need to solve for y. This is the given equation. 2x – 3y + 6 = 0 To isolate the y-term, add 3 y on both sides. 2x + 6 = 3y Reverse the two sides. (This step is optional.) 3y = 2x + 6 Divide both sides by 3. The coefficient of x, 2/3, is the slope and the constant term, 2, is the y-intercept.

Example: Find the slope and y-intercept of the equation. (c) x = -2 (b) y = 3 m = undefined b = none (a) y = ½ x + 5 y = mx + b y = mx + b y = 0x + 3 m = ½ b = 5 m = 0 b = 3

Example: Find the slope and y-intercept of the equation. d) 2x – 3y = 18 m = 2/3 b = - 6 y = mx + b Solve for y. y = 2/3 x - 6 2x – 3y = 18 – 3y = 18 – 2x y = -6 + 2/3 x y = 2/3 x - 6

Your Turn: Find the slope and y-intercept of the equation. 1) 3y + 12 = 6x 2) 3y – x = 12 Solve for y. Solve for y. 3y = 6x – 12 3y = x + 12 y = 2x – 4 y = 1/3 x +4 y = mx + b y = mx + b y = 2x – 4 y = 1/3 x + 4 m = 2 b = –4 m = 1/3 b = 4

Your Turn: 1. y = -x + 8 2. x + 4y = 24 m = b = m = b = -1 8 -1/4 6 -1 undef. none

Example: Write the equation of the line in slope-intercept form. slope = ; y-intercept = 4 y = mx + b Substitute the given values for m and b. y = x + 4 Simplify if necessary.

Example: Write the equation that describes the line in slope-intercept form. slope = –9; y-intercept = y = mx + b Substitute the given values for m and b. y = –9x + Simplify if necessary.

Your Turn: Write the equation that describes the line in slope-intercept form. slope = 3; y-intercept = y = mx + b Substitute the given values for m and b. Simplify if necessary.

Your Turn: Write the equation that describes the line in slope-intercept form. slope = ; y-intercept = –6 y = mx + b Substitute the given values for m and b. Simplify if necessary.

Example: Writing Linear Equations in Slope-Intercept Form Write the equation that describes the line in slope-intercept form. slope = 2; (3, 4) is on the line Step 1 Find the y-intercept. y = mx + b Write the slope-intercept form. 4 = 2(3) + b Substitute 2 for m, 3 for x, and 4 for y. –2 = b 4 = 6 + b –6 –6 Solve for b. Since 6 is added to b, subtract 6 from both sides to undo the addition.

Example: Continued Step 2 Write the equation. y = mx + b y = 2x + (–2) slope = 2; b = - 2 Step 2 Write the equation. y = mx + b Write the slope-intercept form. y = 2x + (–2) Substitute 2 for m, and –2 for b. y = 2x – 2

Your Turn: A line has a slope of 8 and (3, –1) is on the line. Write the equation that describes this line in slope-intercept form. Step 1 Find the y-intercept. y = mx + b Write the slope-intercept form. –1 = 8(3) + b Substitute 8 for m, 3 for x, and –1 for y. –25 = b –1 = 24 + b –24 –24 Solve for b. Since 24 is added to b, subtract 24 from both sides to undo the addition.

Continued Step 2 Write the equation. y = mx + b y = 8x + (–25) slope = 8; b = - 25 Step 2 Write the equation. y = mx + b Write the slope-intercept form. y = 8x + (–25) Substitute 8 for m, and –25 for b. y = 8x – 25

Example: Writing an Equation From a Graph Find the linear equation of the line shown on the graph. y-intercept = 4 y • • Rise = –2 Step 1 Pick two points on the line and determine the slope of the line from the graph. • • m = Run = 5 Step 2 Find the y-intercept from the graph. b = 4 Step 3 Write the linear equation in slope-intercept form (y = mx+ b).

Your Turn: y = –x + 2 y = 2x – 2 Write the equation of the line shown. B. A. y = –x + 2 y = 2x – 2

Example: Write an Equation From Two Points What equation in slope-intercept form represents the line that passes through the points (2, 1) and (5, -8). Step 1 Use the two points to find the slope. Step 2 Use the slope and one of the points to find b. Step 3 Substitute the slope and y-intercept into the slope-intercept form (y = mx + b).

Your Turn: What equation in slope-intercept form represents the line that passes through the points (1, -6) and (-3, 10). Step 1 Use the two points to find the slope. Step 2 Use the slope and one of the points to find b. Step 3 Substitute the slope and y-intercept into the slope-intercept form (y = mx + b).

Your Turn: What equation in slope-intercept form represents the line that passes through the points (3, -2) and (1, -3). Step 1 Use the two points to find the slope. Step 2 Use the slope and one of the points to find b. Step 3 Substitute the slope and y-intercept into the slope-intercept form (y = mx + b).

Joke Time Why shouldn’t you write with a broken pencil? Because it is pointless! What do you call a man with no arms and no legs playing in the leaves? Russell! Which side of a cheetah has the most spots? The outside.

Assignment 5-3 Part 1 Exercises Pg. 332 - 333: #6 – 40 even