Graphs of linear equations and functions: foundations

[Video Placeholder - Introduction to graphing linear equations]

Duration: 5:18

Note: Understanding how to graph linear equations is essential for the SAT Math section. You'll need to interpret graphs, find equations from graphs, and solve problems using graphical representations.

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Graphing linear equations

The graph of a linear equation in two variables (x and y) is a straight line. The standard form is:

y = mx + b

Where:

Graph of y = 2x + 1

Key features of linear graphs

Example 1: Graphing from slope-intercept form

Graph the equation: y = -½x + 3

Step 1: Identify the y-intercept

b = 3 → point at (0, 3)

Step 2: Use the slope to find another point

m = -½ → down 1, right 2
From (0,3), move to (2,2)

Step 3: Draw the line through these points

Try it!

Graph the equation: y = 3x - 2

What is the y-intercept?

What is the slope?

Remember: y = mx + b where b is the y-intercept and m is the slope. Start by plotting the y-intercept, then use the slope to find another point.

Solution:

  1. Plot the y-intercept at (0, -2)
  2. From there, use slope 3 (rise 3, run 1) to plot (1, 1)
  3. Draw a line through these points

Finding equations from graphs

Example 2: Writing equations from graphs

Find the equation of the line shown:

Step 1: Find the y-intercept

The line crosses the y-axis at (0, 1) → b = 1

Step 2: Calculate the slope

Choose two points: (0,1) and (2,0)
Slope (m) = (0-1)/(2-0) = -½

Step 3: Write the equation

y = -½x + 1

Practice: Find the equation

What is the equation of this line?

y = x +

Special cases

Example 3: Horizontal and vertical lines

Horizontal lines have slope 0 (y = constant)

Vertical lines have undefined slope (x = constant)

What are the equations of these lines?

Horizontal line: y = 2 (crosses y-axis at 2)

Vertical line: x = -2 (crosses x-axis at -2)

Up next: Video - Systems of linear equations graphically