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Digital SAT Math

Course: Digital SAT Math > Unit 2

Lesson 4: Graphs of linear equations and functions: foundations
Test prep
Digital SAT Math
Foundations: Algebra
Graphs of linear equations and functions: foundations
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Graphs of linear equations and functions | Lesson

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A guide to graphs of linear equations and functions on the digital SAT

What are graphs of linear equations and functions questions?

Graphs of linear equations and functions questions deal with linear equations and functions and their graphs in the x, y-plane. For example, the graph of y, equals, 2, x, minus, 1 is shown below.
The linear equation y=2x-1 is graphed in the xy-plane. The line trends upward from left to right and passes through the points (0, -1) and (1, 1).
An equation in function notation, f, left parenthesis, x, right parenthesis, equals, 2, x, minus, 1, can also represent this line.
In this lesson, we'll learn to:
  1. Identify features of linear graphs from their equations
  2. Write linear equations based on graphical features
  3. Determine the equations of parallel and perpendicular lines
You can learn anything. Let's do this!

What are the features of lines in the x, y-plane?

Intro to slope

Khan Academy video wrapper
Intro to slopeSee video transcript

Features of lines in the x, y-plane

The slope

The slope of a line describes its direction and steepness.
  • A line that trends upward from left to right has a positive slope.
  • A line that trends downward from left to right has a negative slope.
  • The steeper the line is, the larger the
    of its slope is.
The slope is equal to the ratio of a line's change in y-value to its change in x-value. We can calculate the slope using any two points on the line, left parenthesis, x, start subscript, 1, end subscript, comma, y, start subscript, 1, end subscript, right parenthesis and left parenthesis, x, start subscript, 2, end subscript, comma, y, start subscript, 2, end subscript, right parenthesis:
start text, s, l, o, p, e, end text, equals, start fraction, start text, c, h, a, n, g, e, space, i, n, space, end text, y, divided by, start text, c, h, a, n, g, e, space, i, n, space, end text, x, end fraction, equals, start fraction, y, start subscript, 2, end subscript, minus, y, start subscript, 1, end subscript, divided by, x, start subscript, 2, end subscript, minus, x, start subscript, 1, end subscript, end fraction
Example: Line ell contains the points left parenthesis, minus, 1, comma, 2, right parenthesis and left parenthesis, 4, comma, 12, right parenthesis. What is the slope of line ell ?
A horizontal line has a slope of 0 since all points on the line have the same y-coordinate (so the change in y is 0).
A vertical line has an undefined slope since all points on the line have the same x-coordinate (so the change in x is 0).

The y-intercept

The y-intercept of the line is the point where the line crosses the y-axis. This point always has an x-coordinate of 0. All non-vertical lines have exactly one y-intercept.

The x-intercept

The x-intercept of the line is the point where the line crosses the x-axis. This point always has a y-coordinate of 0. All non-horizontal lines have exactly one x-intercept.

Try it!

Try: identify the features of a graph
A line is graphed in the xy-plane. The line trends downward from left to right and passes through the points (0, 3) and (2, 0).
The graph of line m is shown above.
The y-intercept of the line is located at left parenthesis, 0, comma
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
right parenthesis.
The x-intercept of the line is located at left parenthesis
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
comma, 0, right parenthesis.
We can use the intercepts to calculate the slope of the line. The slope of the line is
  • Your answer should be
  • an integer, like 6
  • a proper fraction, like 1, slash, 2 or 6, slash, 10
  • an improper fraction, like 10, slash, 7 or 14, slash, 8
  • an exact decimal, like 0, point, 75
.

How do I tell the features of lines from linear equations?

Converting to slope-intercept form

Khan Academy video wrapper
Converting to slope-intercept formSee video transcript

How do I interpret an equation in slope-intercept form?

Lines in the x, y-plane are visual representations of linear equations. The slope-intercept form of a linear equation, y, equals, m, x, plus, b, tells us both the slope and the y-intercept of the line:
  • The slope is equal to m.
  • The y-intercept is equal to b.
For example, the graph of y, equals, start color #7854ab, 3, end color #7854ab, x, start color #ca337c, minus, 7, end color #ca337c has a slope of start color #7854ab, 3, end color #7854ab and a y-intercept of start color #ca337c, minus, 7, end color #ca337c.
Because the slope-intercept form shows us the features of the line outright, it's useful to rewrite any linear equation representing a line in slope-intercept form.
Example: What is the slope of the graph of 3, x, plus, 4, y, equals, 12 ?

Try it!

Try: identify slope and intercept from a linear equation
2, x, plus, y, equals, 3
To rewrite the above equation in slope-intercept form, we can isolate y by
both sides of the equation.
When 2, x, plus, y, equals, 3 is graphed in the x, y-plane:
  • The slope of the line is
    • Your answer should be
    • an integer, like 6
    • a simplified proper fraction, like 3, slash, 5
    • a simplified improper fraction, like 7, slash, 4
    • a mixed number, like 1, space, 3, slash, 4
    • an exact decimal, like 0, point, 75
    • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
    .
  • The y-intercept of the line is
    • Your answer should be
    • an integer, like 6
    • a simplified proper fraction, like 3, slash, 5
    • a simplified improper fraction, like 7, slash, 4
    • a mixed number, like 1, space, 3, slash, 4
    • an exact decimal, like 0, point, 75
    • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
    .

How do I write linear equations based on slopes and points?

Slope-intercept equation from two points

Khan Academy video wrapper
Slope-intercept equation from two pointsSee video transcript

What information do I need to write a linear equation?

We can write the equation of a line as long as we know either of the following:
  • The slope of the line and a point on the line
  • Two points on the line
In both cases, we'll be using the information provided to find the missing values in y, equals, m, x, plus, b.

The slope and a point

When we're given the slope and a point, we have values for x, y, and m in the equation y, equals, m, x, plus, b, and we just need to plug in the values and solve for the y-intercept b.
Example: If line a has a slope of 2 and passes through the point left parenthesis, 1, comma, 3, right parenthesis, what is the equation of line a ?
Note: if the given point is the y-intercept, then we just need to plug in the slope for m and the y-intercept for b. No calculation needed!

Two points

When we're given two points, we must first calculate the slope using the two points, then plug in the values of x, y, and m into y, equals, m, x, plus, b to find b.
Example: Line b passes through the points left parenthesis, minus, 2, comma, 4, right parenthesis and left parenthesis, 1, comma, minus, 5, right parenthesis. What is the equation of line b ?

Try it!

TRY: WRITE the equation of a line
A line in the xy-plane trends upward from left to right and passes through the points (0, -4) and (2, 2).
The line shown above passes through the points left parenthesis, 0, comma
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
right parenthesis and left parenthesis, 2, comma, 2, right parenthesis.
The slope of the line is
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
.
The equation of the line in slope-intercept form is:
y, equals

How do I write equations of parallel and perpendicular lines?

Parallel & perpendicular lines from graph

Khan Academy video wrapper
Parallel & perpendicular lines from graphSee video transcript

What are the features of parallel and perpendicular lines?

In the x, y-plane, lines with different slopes will intersect exactly once.
Parallel lines in the x, y-plane have the same slope. Parallel lines do not intersect unless they also completely overlap (i.e., are the same line).
Perpendicular lines in the x, y-plane have slopes that are
of each other. Perpendicular lines form 90, degrees angles.
The graph below shows lines ell, m, and n.
  • Line start color #7854ab, ell, end color #7854ab has a slope of 2.
  • Line start color #ca337c, m, end color #ca337c also has a slope of 2. It is parallel to line start color #7854ab, ell, end color #7854ab.
  • Line start color #208170, n, end color #208170 has a slope of minus, start fraction, 1, divided by, 2, end fraction. It is perpendicular to both lines start color #7854ab, ell, end color #7854ab and start color #ca337c, m, end color #ca337c.
Lines l, m, and n are graphed in the xy-plane. Line l has a slope of 2 and a y-intercept of -1. Line m has a slope of 2 and a y-intercept of -4. Line n has a slope of -1/2 and a y-intercept of 1. Lines l and m are parallel. Line n is perpendicular to lines l and m.
This means we can write the equation of a parallel or perpendicular line based on a slope relationship and a point on the line.

Let's look at some examples!

Lines p and q are graphed in the x, y-plane. Line p is represented by the equation y, equals, minus, 2, x, minus, 5. If line q is parallel to line p and passes through the point left parenthesis, 0, comma, 4, right parenthesis, what is the equation of line q ?
Line ell is represented by the equation y, equals, minus, 3, x, plus, 2. What is the equation of a line that is perpendicular to line ell and intersects line ell at left parenthesis, minus, 3, comma, 11, right parenthesis ?

Try it!

TRY: identify the features of parallel and perpendicular lines
A line in the xy-plane trends downward from left to right and contains the points (0, 3) and (3, 1).
The line shown in the graph above has a slope of
  • Your answer should be
  • an integer, like 6
  • a proper fraction, like 1, slash, 2 or 6, slash, 10
  • an improper fraction, like 10, slash, 7 or 14, slash, 8
and a y-intercept of
  • Your answer should be
  • an integer, like 6
  • a proper fraction, like 1, slash, 2 or 6, slash, 10
  • an improper fraction, like 10, slash, 7 or 14, slash, 8
.
The slope of a line parallel to the line above must have a slope of
  • Your answer should be
  • an integer, like 6
  • a proper fraction, like 1, slash, 2 or 6, slash, 10
  • an improper fraction, like 10, slash, 7 or 14, slash, 8
.
The slope of a line perpendicular to the line above must have a slope of
  • Your answer should be
  • an integer, like 6
  • a proper fraction, like 1, slash, 2 or 6, slash, 10
  • an improper fraction, like 10, slash, 7 or 14, slash, 8
  • an exact decimal, like 0, point, 75
.
The slope-intercept form equation of a perpendicular line that intersects the line shown above at left parenthesis, 0, comma, 3, right parenthesis is:
y, equals

Your turn!

Practice: match an equation and its graph
Which of the following is the graph of the equation y, equals, minus, start fraction, 1, divided by, 2, end fraction, x, plus, 3 in the x, y-plane?
Choose 1 answer:

Practice: write an equation based two points
Line ell in the x, y-plane passes through the points left parenthesis, minus, 1, comma, minus, 2, right parenthesis and left parenthesis, 2, comma, 7, right parenthesis. Which of the following equations describes line ell ?
Choose 1 answer:

Practice: find a point on a perpendicular line
In the x, y-plane, line ell has a y-intercept of minus, 7 and is perpendicular to the line with equation y, equals, minus, start fraction, 1, divided by, 4, end fraction, x. If the point left parenthesis, 5, comma, c, right parenthesis is on line ell, what is the value of c ?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text

Things to remember

start text, s, l, o, p, e, end text, equals, start fraction, start text, c, h, a, n, g, e, space, i, n, space, end text, y, divided by, start text, c, h, a, n, g, e, space, i, n, space, end text, x, end fraction, equals, start fraction, y, start subscript, 2, end subscript, minus, y, start subscript, 1, end subscript, divided by, x, start subscript, 2, end subscript, minus, x, start subscript, 1, end subscript, end fraction
The slope-intercept form of a linear equation, y, equals, m, x, plus, b, tells us both the slope and the y-intercept of the line:
  • The slope is equal to m.
  • The y-intercept is equal to b.
We can write the equation of a line as long as we know either of the following:
  • The slope of the line and a point on the line
  • Two points on the line
Parallel lines in the x, y-plane have the same slope.
Perpendicular lines in the x, y-plane have slopes that are negative reciprocals of each other.

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