Problem 63 - Introductory Algebra for College Students

Question

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. If a parabola has only one \(x\) -intercept, then the \(x\) -intercept is also the vertex.

Step-by-Step Solution

Answer

The statement is True

Step 1: Understanding the Problem

First, clearly understand the problem. The statement to be evaluated for truthfulness is: 'If a parabola has only one x-intercept, then the x-intercept is also the vertex.' This means if a parabola intersects the x-axis at only one point, that point is also the vertex of the parabola.

Step 2: Evaluating the Statement

Next, evaluate the truthfulness of the statement. A parabola has only one x-intercept when the curve touches the x-axis at only one point. This happens when the parabola's vertex is on the x-axis, since the vertex is the point at the maximum or minimum of the parabola, and any movement up or down from the vertex would move the parabola away from the x-axis. Therefore, in such cases, the x-intercept is the vertex of the parabola.

Step 3: Final Answer

Comparing the given statement with the conclusion from step 2, it's clear that the statement 'If a parabola has only one x-intercept, then the x-intercept is also the vertex' is indeed True

Key Concepts

x-intercept

The x-intercept is the point where a graph crosses the x-axis, meaning the y-coordinate is zero at this point. For a quadratic equation, which forms a parabola, finding the x-intercepts is crucial, as they tell us where the graph intersects the x-axis. Parabolas can have two, one, or no x-intercepts.

• Two x-intercepts: The parabola crosses the x-axis at two distinct points.
• One x-intercept: This occurs when the parabola just touches the x-axis, indicating that the vertex of the parabola is on the x-axis. Hence, the x-intercept and the vertex are the same.
• No x-intercepts: The entire parabola is above or below the x-axis, without intersecting it.

Understanding the x-intercepts helps in graph sketching and predicting the behavior of quadratic functions.

vertex

The vertex of a parabola is a crucial point depicting its maximum or minimum value. For a standard quadratic equation written as \(y = ax^2 + bx + c\), the vertex can be found using the formula \(x = -\frac{b}{2a}\). Once you have the x-value of the vertex, you can find the y-coordinate by plugging it back into the original quadratic equation. When a parabola has exactly one x-intercept, this special point is also its vertex, meaning the vertex is directly on the x-axis. This is why the parabola touches the x-axis at just that one point and the orientation (either upwards or downwards) determines if the vertex is a minimum or maximum.

quadratic functions

Quadratic functions are polynomial functions of degree two, generally represented by the equation \(y = ax^2 + bx + c\). These functions form a characteristic U-shaped graph called a parabola. Parabolas can open upwards or downwards based on the sign of the leading coefficient \(a\):

• Upward-opening: If \(a > 0\), the parabola opens upwards, and the vertex is the minimum point.
• Downward-opening: If \(a < 0\), the parabola opens downwards, and the vertex is the maximum point.

Understanding the properties of quadratic functions, such as the x-intercepts and vertex, allows you to analyze and sketch the graph effectively. Quadratic functions are widely used in various fields such as physics, engineering, finance, and economics for modeling and analysis due to their simple yet versatile nature.