Question

Multiple Choice The graph of a function \(y=f(x)\) can have more than one of which type of intercept? (a) \(x\) -intercept (b) \(y\) -intercept (c) both (d) neither

Short answer

(a) x-intercept

Step-by-step solution

Understanding Intercepts

Intercepts are points where the graph of a function crosses the axes. An x-intercept is where the graph crosses the x-axis (i.e., where the value of y is zero), and a y-intercept is where the graph crosses the y-axis (i.e., where the value of x is zero).

Analyzing the x-intercepts

A function can have multiple x-intercepts. For example, the function \[y = x^2 - 1\] has two x-intercepts at x = 1 and x = -1.

Analyzing the y-intercept

A function typically has at most one y-intercept because it can only cross the y-axis at one point. For example, the function \[y = x^2 - 1\] has a y-intercept at y = -1.

Evaluating the choices

From the analysis, we can see that a function can have more than one x-intercept but usually only one y-intercept. So, the correct choice is the one indicating multiple x-intercepts.

Key concepts

x-intercept

The x-intercept is the point where a function's graph crosses the x-axis. This is the location where the y-value equals zero.

For instance, in the function \(y = x^2 - 1\), the graph touches the x-axis at two points: x = 1 and x = -1.

Solving for the x-intercepts involves setting y to zero and solving for x. Here's an example:

\[0 = x^2 - 1 \]
By rearranging and solving for x, we get:

\[ x^2 = 1 \]
\[ x = 1 \] and \[ x = -1 \]

This means the function has two x-intercepts at x = 1 and x = -1. Functions can have multiple x-intercepts, depending on their shape and nature.

y-intercept

The y-intercept is where a function's graph crosses the y-axis. This happens where the x-value is zero.

Unlike x-intercepts, a function typically has just one y-intercept.

For the same function, \(y = x^2 - 1\), substituting x = 0 gives:

\[ y = 0^2 - 1 \]
\[ y = -1 \]

Hence, the y-intercept is at y = -1.

Because the y-axis is a vertical line at x = 0, the graph can usually only cross it at one point.

Some specialized functions can have more than one y-intercept, but this is rare.

function graph analysis

Graph analysis involves interpreting various features and behaviors of a function's graph.

Key points to look at include:

• Intercepts: Both x and y-intercepts show where a function crosses the axes.


• Slope: Indicates how steep the graph is and the direction it goes.


• Shape: Helps determine if the function is linear, quadratic, exponential, etc.

By examining these details, students can gain a better understanding of the function and its overall behavior.

For instance, the function \(y = x^2 - 1\) has a parabolic shape, with x-intercepts at x = 1 and x = -1, and a y-intercept at y = -1.

This insight allows us to visualize and predict how changes in the function will affect its graph.

multiple intercepts

A function can have more than one x-intercept, but usually only one y-intercept.

For example, the quadratic function \(y = x^2 - 1\) has two x-intercepts, indicating it crosses the x-axis at two points.

However, it has just one y-intercept.

Functions like polynomials often have multiple x-intercepts due to their varying shapes.

Intercepts are important for graphing and understanding where a function changes direction or value.

By knowing where these points are, students can accurately sketch and analyze the behavior of different functions.