Intercept Form
Key Concepts
- Graph a linear equation
- Write an equation from a graph
- Understand slope intercept form
- Interpret slope and y- Intercept
Slope – Intercept Form
Slope
The slope of a line is the ratio of the amount that y increases as x increases some amount. Slope tells you how steep a line is or how much y increases as x increases.
The slope is constant (the same) anywhere on the line.
Slope formula
m = rise / run = y2−y1 / x2−x1
Real-life example of slope
Slope-Intercept Form
The meaning of slope-intercept form is the equation of a straight line in the form y = mx + b, where m is the slope of the line and b is its y-intercept.
y = mx + b
Graph a linear equation
Linear equation
A linear equation is an algebraic equation of the form y=mx+b.
Involving only a constant and a first-order (linear) term, where m is the slope and b is the y-intercept. Occasionally, the above is called a “linear equation of two variables,” where y and x are the variables.
Example 1:
What is the graph of y = 3x+1?
Sol:
Step 1:
Identify the y-intercept in the equation.
The y-intercept is 1, So plot the points (0, 1).
Step 2:
Use the slope to plot a second point.
m = 3 = vertical change / horizontal change
Start at (0, 1), move 3 units up and 1 unit to the right to locate a second point.
Plot the points (1, 4).
Step 3:
Draw a line through the points.
Write an equation from a graph
Example:
What is the equation of the line in slope-intercept form?
Sol:
Step 1:
Find the slope between two points on the line.
The line passes through (-3, 3) and (3, -1).
Slope =
y2−y1 / x2−x1
= (−1)−3 / 3− (−3)
=−4 / 6
= –2 / 3
Step 2:
Find the y–intercept.
The line intersects the y-axis at (-3, 3) so the y-intercept is 1.
Step 3:
Write the equation in the form y=mx+b.
Substitute −2 / 3 for m and 1 for b.
The equation of the line in the slope-intercept form is,
y = 2 / 3 x + 1
Understand Slope–Intercept form
Example:
Write the equation of the line that passes through the given points (2, 2) and (3, 4).
Sol:
Step 1:
Find the slope of the line.
m = y2−y1 / x2−x1
m = 4 − 2 / 3 − 2
m = 2
Step 2:
Use the slope and one point to find the y-intercept.
y = mx +b
4 = 2(3) +b Substitute 2 for m. (3, 4) for (x, y)
4 = 6 +b Simplify
-2 = b
Step 3:
Use the slope and y-Intercept to write the equation.
y = mx +b
y = 2x -2 Substitute 2 for m and -2 for b.
The equation in slope–intercept form of the line that passes through (2, 2) and (3, 4) is y = 2x -2.
Interpret Slope and y-Intercept
Example:
The rent charged for space in an office building as a linear relationship related to the size of the space rented. Write an equation in slope-intercept form for rent at west main street office rentals
Sol:
Step 1:
Create a table or identify ordered pairs from the problem.
(600, 750) and (900, 1150)
Step 2:
Find the slope using the slope formula,
m = y2−y1 / x2−x1
m= 1150−750 / 900−600
m= 400 / 300
m= 4 / 3
Step 3:
Write the form y= mx+b.
Step 4:
Replace m with the value you found.
Y= 4 / 3 x+b
Step 5:
Plug-in one of the points you already know and solve for b using the inverse operation.
750 = 4 / 3 (600)+b
750 = 800 +b
-50=b
Step 6:
Replace m and b with the values.
y= 4 / 3 x+(-50)
Y= 4 / 3 x -50.
Exercise
- Sketch the graph of the equation.
y = 2x -5
- Identify the slope and y-intercept of the line for the equation.
Y = – 5x – 3/4
- Write the equation of the line that passes through the given points (0, 1) and (2, 2).
- Write the equation of each line in slope–intercept form.
- Jordan will hike the trail shown at a rate of 4 mi/h. Write a linear equation to represent the distance Jordan still has to walk after x hours. What does the y-intercept of the equation represent?
- Write the equation of the line that passes through the given points (5, 4) and (-1, 6).
- Write the equation of the line in slope–intercept form.
- Graph the following line:
- Y = x + 7.
- Find the equation of the line graphed below.
- Shriya read a book cover to cover in a single session, at a constant rate. After reading for 1.5 hours, she had 402 out of the total 480 pages left to read. Let y represent the number of pages left to read after x hours. Complete the equation for the relationship between the number of pages left and the number of hours. y=___________.
Concept Map
What have we learned
- Understand Slope –Intercept Form
- Graph a linear equation
- Understand how to write an equation from a graph
- Interpret slope and y- Intercept
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