Lesson 30: Graphing Functions

New Concept

A linear function is a function whose graph is a line. A linear function can be written in the form $f(x) = mx + b$, where $m$ and $b$ are real numbers.

What’s next

You're at the start of your algebra journey! Next, we'll dive into graphing functions using tables, matching equations to graphs, and identifying their domains.

Property

A linear equation is an equation whose graph is a line. A linear function is a function whose graph is a line and can be written in the form $f(x) = mx + b$, where $m$ and $b$ are real numbers.

Examples

• The equation $y = x + 3$ is linear. Plotting points like $(0, 3)$ and $(1, 4)$ will form a straight line on a graph.
• The car wash fundraiser follows the rule $f(x) = 5x - 4$. Since the graph is a straight line, it represents a linear function.
• An equation like $y = -x + 10$ is linear because for every one unit you move right, you move one unit down, creating a straight line.

Explanation

Think of a linear function as a perfectly straight path! For every step you take sideways (the x-value), you go up or down by the exact same amount (the slope). This consistency creates a straight line, not a wild curve. We can map out this predictable journey by plotting a few points from a table and connecting them.

Property

If a vertical line intersects the graph at more than one point, then the graph is not a function.

Examples

• A parabola like $y = x^2$ passes the test. Any vertical line drawn will only intersect the curve at a single point, so it is a function.
• A straight line like $y = x$ also passes the test. Any vertical line will only cross it once, confirming it is a function.
• The graph of a circle, like $x^2 + y^2 = 1$, fails the test. A vertical line drawn through its center would hit both the top and bottom of the circle.

Explanation

Imagine scanning a vertical laser beam across your graph from left to right. If that beam ever hits your graphed line in more than one place at the same time, it fails the test! A true function gives only one output (y) for each input (x), so the laser should only ever touch one point at a time.

Property

You can use a table of ordered pairs to graph an equation.

Examples

• To graph $y = x^2$, a table might include $(-2, 4)$, $(-1, 1)$, $(0, 0)$, $(1, 1)$, and $(2, 4)$. Plotting these points reveals a U-shaped parabola.
• For the linear equation $y = 5x - 4$, a table could have points $(0, -4)$, $(1, 1)$, and $(2, 6)$. Plotting these points creates a straight line.
• To see if a graph matches $f(x) = \frac{1}{3}x + 4$, you can check if points from its table, like $(0,4)$ and $(3,5)$, lie on the graphed line.

Explanation

This is like playing connect-the-dots, but with math! First, you choose a few input values for 'x' and plug them into the equation to find their partner 'y' values. Organize these (x, y) pairs in a table. Then, plot each point on the graph and connect them to reveal the function's shape, whether it's a line or a curve.