Slope Intercept Form

Slope Intercept Form

 $y=mx$  and  $y=mx+b$  

Let's recap what we already know...

• Slope is a measure of change.

• Slope is a measure of steepness.

• Slope can be positive, negative, zero, or undefined.

Slope from a graph

 $m=\frac{vertical\ change}{horizontal\ change}=\frac{rise}{run}$  

Use slope triangles to find the vertical change as a ratio to the horizontal change.

Multiple Choice

Find the slope from the graph.

• $m=\frac{3}{2}$

• $m=-\frac{2}{3}$

• $m=\frac{2}{3}$

• $m=-\frac{3}{2}$

Slope from coordinate points

• Watch your +/- signs

• Always simplify your fraction

Create an xy chart

Multiple Choice

Find the slope of a line that passes through the points (0, 3) and (3, 8).

• $m=\frac{3}{5}$

• $m=\frac{8}{0}$

• $m=-\frac{5}{3}$

• $m=\frac{5}{3}$

Slope from a table

• Find the change from one value to the next.

• Remember, it's y/x

Multiple Choice

Find the slope from the table of values.

•  $m=\frac{1}{2}$  

•  $m=2$  

•  $m=\frac{1}{5}$  

•  $m=10$  

Intercepts are the coordinate points where a line crosses the x and y axis.

• Intercepts are coordinate points, not a single number.

Multiple Choice

Name the x-intercept from the graph.

• (-2, 0)

• (0, -2)

• (4, 0)

• (0, 4)

Multiple Choice

Name the y-intercept from the graph.

• (-2, 0)

• (0, -2)

• (4, 0)

• (0, 4)

All straight lines can be written as a linear equation using a point on the line and the slope of the line.

Slope-Intercept Form of an equation

• The equation of a line can be written in the form y = mx + b, where m is the slope and b is the y-value portion of the y-intercept.

• The y represents the output or dependent variable. The x is the input or independent variable.

 $y\ =mx$  

• Graphs that pass through the origin can be written in the form y = mx, where m is the slope.  The y-intercept is (0, 0) and does not need to be included in the equation (y = 1/2x + 0).

• Also known as the constant of proportionality or a unit rate.

• y represents the output or dependent variable.

• x represents the input or independent variable.

• In this example, the number of batches of cookies, y, is dependent on the cups of flower, x.

 $y=mx+b$  

• Other situations require a "starting point."  The y-intercept is that starting point.

• In this example, the baby's overall length can be tracked using the equation  $y=\frac{1}{4}x+20$ , where the slope, or rate of change, is 1/4.  

• The baby grows 1 inch every 4 weeks.  In 16 weeks the baby will be 24 inches long, not 4 inches, because the starting point was 20 inches.

A tree was 8 feet tall when planted and grows at a rate of 1 foot per year.

• The slope, or rate of change is 1.

• The starting point is the y-intercept (0, 8).

Other situations

• Cell phone: Total cost = $35 per line + $100 15GB data plan (y = 35x + 100)

• Service Call: $75 per hour plus $50 service fee (y = 75x + 50)

• Leaky Pool: Pool filled with 690,000 gallons of water leaking at a rate of 5 gallons per minute (y = -5x + 690,000)

• Travel Ball Cost: $450 + $50 each tournament (y = 450 + 50x)

Open Ended

Describe another situation where the starting point is something other than the origin. Write an equation to represent your situation.

Writing an equation in slope-intercept form

As previously mentioned, all you need to write an equation in slope-intercept form (y = mx + b) is the slope, or rate of change, and a point on the line. The y-intercept is one point on the line. Sometimes, the y-intercept is the origin (0, 0).

Write the equation of a line with a slope of 3/4 and a y-intercept of (0, 6).

•  $y=mx+b$  

• Substitute the slope for the m and y portion of the y-intercept for b.

•  $y=\frac{3}{4}x+6$  

Write the equation of a line with a slope of 4 and a y-intercept of (0, -3)

•  $y=mx+b$  

• Substitute the slope for the m and y portion of the y-intercept for b.

• If the y-intercept is negative, the equation will have a minus sign.

•  $y=4x-3$  

Multiple Choice

Write the equation of a line with a slope of 2 and a y-intercept of (0, 8).

• $y=8x+2$

• $y=\frac{2}{8}x$

• $y=2x+08$

• $y=2x+8$

Multiple Choice

Write the equation of a line with a slope of -5/3 and y-intercept of (0, 0)

• $y=-\frac{5}{3}x$

• $y=0x+5$

• $y=\frac{3}{5}x-5$

Multiple Choice

Write the equation of a line that has a slope of 1/2 and a y-intercept of (0, -12).

• $y=-\frac{1}{2}x+12$

• $y=\frac{1}{2}x-12$

• $y=-2x-12$

Identify the slope and y-intercept

•  $y=\frac{2}{3}x-9$  

• The slope is the number with the variable.  The y-intercept is the constant term, or the number that does not have a variable.

• Remember, the y-intercept is a coordinate point, not just a number.  It's the point where the graph crosses the y-axis and will always be (0, b).

• slope = 2/3

• y-intercept is (0, -9)

Identify the slope and y-intercept

•  $y=-5x+6$  

• slope = -5

• y-intercept is (0, 6)

• If the coefficient is negative, the slope is negative.

Identify the slope and y-intercept

•  $y=2.6x$  

• If there is no constant term, the y-intercept is the origin (0, 0).

• slope = 2.6

• y-intercept is (0, 0)

• It's not necessary to write  $y=2.6x+0$  . The + 0 does not add any value to the equation.

Multiple Choice

Identify the slope and y-intercept in the equation,

 $y=-3x+1$  

• m = -3
  y-int: (0, 1)

• m = 3
  y-int: (0, 1)

• m = 1
  y-int: (-3, 0)

• m = 1
  y-int: (1, 0)

Multiple Choice

What is the slope in the equation

 $y=-\frac{2}{3}x-2$  

• -2

• -2/3

• 23

• (0, -2)