X-Intercept

Definition

The x-intercept of a graph is the point where the graph intersects the x-axis, indicating the value of x when the function's output or y-value is zero. It represents the horizontal location where a curve or line crosses the x-axis.

5 Must Know Facts For Your Next Test

1. The x-intercept is a key feature of linear, polynomial, rational, and logarithmic functions, as it helps determine the behavior and characteristics of the graph.
2. For linear functions, the x-intercept can be found by setting the equation equal to zero and solving for x.
3. In polynomial functions, the x-intercepts correspond to the zeros or roots of the function, which can be found using factoring, the quadratic formula, or other polynomial root-finding methods.
4. Rational functions often have asymptotes that intersect the x-axis, and these points of intersection represent the x-intercepts of the function.
5. The x-intercept of a logarithmic function can be found by setting the function equal to zero and solving for the input variable.

Review Questions

• Explain how to find the x-intercept of a linear function.
  • To find the x-intercept of a linear function, you set the function equal to zero and solve for the x-variable. This gives you the x-value where the graph of the linear function intersects the x-axis. For example, if the linear function is $y = 2x - 4$, you would set $2x - 4 = 0$ and solve for $x$, which gives you $x = 2$. Therefore, the x-intercept of this linear function is (2, 0).
• Describe how the x-intercepts of a polynomial function relate to the roots or zeros of the function.
  • The x-intercepts of a polynomial function correspond to the roots or zeros of the function, where the function's output is equal to zero. These x-intercepts can be found by factoring the polynomial equation, using the quadratic formula, or employing other polynomial root-finding methods. The number of x-intercepts a polynomial function has is equal to the degree of the polynomial, as each root represents a point where the graph crosses the x-axis.
• Explain the relationship between the x-intercepts of a rational function and its asymptotes.
  • Rational functions often have vertical asymptotes, which are vertical lines that the graph of the function approaches but never touches. The x-intercepts of a rational function occur at the points where the graph intersects the x-axis, and these x-intercepts can also correspond to the points where the vertical asymptotes of the function intersect the x-axis. Understanding the connection between the x-intercepts and asymptotes of a rational function is crucial for accurately sketching the graph and analyzing the function's behavior.

Related terms

Y-Intercept: The y-intercept of a graph is the point where the graph intersects the y-axis, indicating the value of y when the function's input or x-value is zero. It represents the vertical location where a curve or line crosses the y-axis.

Slope:

The slope of a line is the measure of its steepness, indicating the rate of change between the x and y values. It represents the ratio of the change in y to the change in x.

Asymptote:

An asymptote is a line that a curve or graph approaches but never touches. It represents the limiting value that a function approaches as the input or output variable approaches a certain value.