Lesson 2: Graphs of Functions

New Concept

A function's graph is a visual map of its input-output pairs. Each point $(x, y)$ on the graph corresponds to an input, $x$, and its unique output, $y = f(x)$. This allows us to analyze function behavior and solve equations visually.

What’s next

Next, you'll put this concept into action with interactive examples on reading graphs and constructing your own from function rules.

Property

The point $(a, b)$ lies on the graph of the function $f$ if and only if $f(a) = b$. Each point on the graph of the function $f$ has coordinates $(x, f(x))$ for some value of $x$.

Examples

• If $f(5) = 8$, the point $(5, 8)$ is on the graph of the function $f$.
• If the point $(-1, 4)$ is on the graph of a function $g$, it means that $g(-1) = 4$.
• The coordinates of any point on the graph of $h$ can be written in the form $(x, h(x))$ for some input $x$.

Explanation

A function's graph is a picture of all its input-output pairs. The horizontal position (x-coordinate) is the input, and the vertical position (y-coordinate) is the corresponding output. It's a visual map of the function's behavior.

Property

To find a function's value $f(x)$ from its graph, locate the input $x$ on the horizontal axis, move vertically to the graph, then move horizontally to find the output value on the vertical axis.

Examples

• To find $f(3)$, find $x=3$ on the horizontal axis. If the point on the graph above it is $(3, -2)$, then $f(3) = -2$.
• To find the value(s) of $x$ for which $f(x) = 5$, find $y=5$ on the vertical axis. If the graph has points at $(-1, 5)$ and $(4, 5)$, the solutions are $x=-1$ and $x=4$.
• The highest point on a graph shows the function's maximum value. If the highest point is $(2, 9)$, the maximum value is $9$.

Explanation

Use the graph as a lookup tool. Start on the x-axis for an input to find the output on the y-axis. Or, start on the y-axis for an output to find the input(s) on the x-axis.

Property

We can construct a graph for a function described by an equation by plotting points whose coordinates satisfy the equation. We choose several convenient values for $x$ and evaluate the function to find the corresponding $f(x)$ values.

Examples

• To graph $f(x) = x^2 + 1$, we can choose $x=2$. We find $f(2) = 2^2 + 1 = 5$, so we plot the point $(2, 5)$.
• For the same function, $f(x) = x^2 + 1$, we find $f(0) = 0^2 + 1 = 1$. This gives us the y-intercept at $(0, 1)$.
• After calculating points like $(-2, 5)$, $(0, 1)$, and $(2, 5)$, connecting them reveals the U-shape of the parabola.

Explanation

To draw a function, create a table of values. Pick several inputs ($x$), calculate their outputs ($f(x)$) using the function's rule, and plot each $(x, f(x))$ coordinate pair. Then, connect the dots with a smooth curve.

Property

A graph represents a function if and only if every vertical line intersects the graph in at most one point.

Examples

• A circle fails the vertical line test because a vertical line can cross it twice. Thus, a circle is not the graph of a function.
• A parabola opening upwards or downwards passes the test, as any vertical line hits it only once. It is the graph of a function.
• A sideways 'S' curve might fail the test if a vertical line can pass through it at three different points.

Explanation

This is a quick visual check. If you can draw a single vertical line that hits the graph more than once, it is not a function. This is because one input cannot have multiple outputs.

Property

To solve an equation such as $f(x) = k$ graphically, sketch the graph of $y=f(x)$ and the horizontal line $y=k$. The $x$-coordinates of the points where the graphs intersect are the solutions of the equation.

Examples

• To solve the equation $x^2 - 3 = 1$, graph $y=x^2-3$ and the horizontal line $y=1$. They intersect at $x=-2$ and $x=2$, which are the solutions.
• To solve $\frac{12}{x} = 4$, graph $y = \frac{12}{x}$ and $y=4$. The single intersection point is at $(3, 4)$, so the solution is $x=3$.
• If the graph of $g(x)$ and the line $y=10$ never cross, the equation $g(x)=10$ has no solution.

Explanation

Think of this as finding where two graphs meet. The graph of $f(x)$ shows all possible outputs, and the line $y=k$ highlights one specific output. The x-values of the intersection points are the inputs that produce that output.