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Chapter 2: Problem 88

In Exercises \(87-90\), determine whether each statement is true or false. A quadratic function must have an \(x\) -intercept.

Short Answer

Expert verified
False, a quadratic function might not have an \(x\)-intercept if its discriminant is less than zero.

Step by step solution

01

Understanding the Quadratic Function

A quadratic function is generally given by the form \( f(x) = ax^2 + bx + c \), where \( a, b, \) and \( c \) are constants and \( a eq 0 \). The graph of a quadratic function is a parabola.
02

Identifying the Condition for X-Intercepts

To determine if a quadratic function has an \(x\)-intercept, we need to understand when the function equals zero, i.e., solving \( ax^2 + bx + c = 0 \). The solutions to this equation give the \(x\)-intercepts, if they exist.
03

Working with the Discriminant

The quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) reveals that the discriminant \( b^2 - 4ac \) determines the nature of the roots. If \( b^2 - 4ac > 0 \), there are two real roots; if \( b^2 - 4ac = 0 \), there is one real root; if \( b^2 - 4ac < 0 \), there are no real roots.
04

Evaluating Real Life Examples

Consider \( f(x) = x^2 + 4 \). Solving \( x^2 + 4 = 0 \), we find the discriminant is \(-4\), which is less than zero. Thus, this function has no real roots and no \( x \)-intercepts.
05

Conclusion

Since it is possible for a quadratic function to have a negative discriminant and thus no real roots or \( x \)-intercepts, the statement that a quadratic function must have an \(x\)-intercept is false.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

X-intercepts
An important characteristic of a quadratic function is the x-intercept. These are the points where the graph of the function crosses or touches the x-axis. To find x-intercepts, you need to solve the quadratic equation set to zero, expressed as \( ax^2 + bx + c = 0 \).
  • If the equation has real solutions, those solutions are the x-intercepts.
  • There can be two, one, or no x-intercepts depending on the discriminant, which we'll discuss shortly.
Understanding the nature of x-intercepts helps in sketching the graph of the quadratic function and determining its roots.
Discriminant
The discriminant is a vital part of the quadratic formula. It tells us about the number and nature of the roots of a quadratic equation. The discriminant is given by \( b^2 - 4ac \). Here's what the discriminant indicates:
  • If \( b^2 - 4ac > 0 \), there are two distinct real roots, meaning two distinct x-intercepts.
  • If \( b^2 - 4ac = 0 \), there is one real root, corresponding to one x-intercept where the parabola touches the x-axis.
  • If \( b^2 - 4ac < 0 \), there are no real roots, and thus no x-intercepts because the parabola does not touch the x-axis at all.
This simple calculation can quickly help you determine the possible number of x-intercepts of any quadratic function.
Quadratic Formula
The quadratic formula provides an essential method for finding the roots or x-intercepts of any quadratic equation. The formula is expressed as:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]Using this formula, you can accurately find the x-intercepts as it utilizes the discriminant for computation.
  • The "\(+/-\)" sign indicates that there might be two solutions, which correspond to the two potential x-intercepts in most cases.
  • This formula is very useful, especially when the quadratic equation is difficult to factorize.
Regardless of how complex a quadratic equation appears, the quadratic formula remains a powerful tool to identify and verify the roots efficiently.

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Most popular questions from this chapter

In Exercises \(45-54,\) find the vertex of the parabola associated with each quadratic function. $$f(x)=-3.2 x^{2}+0.8 x-0.14$$

Write a rational function that has vertical asymptotes at \(x=-3\) and \(x=1\) and a horizontal asymptote at \(y=4\)

Graph the rational functions. Locate any asymptotes on the graph. $$f(x)=\frac{1-9 x^{2}}{\left(1-4 x^{2}\right)^{3}}$$

In Exercises \(55-66\), find the quadratic function that has the given vertex and goes through the given point. $$\text { vertex: }\left(\frac{1}{2},-\frac{3}{4}\right) \quad \text { point: }:\left(\frac{3}{4}, 0\right)$$

In Exercises \(55-66\), find the quadratic function that has the given vertex and goes through the given point. vertex: (0,-2)\(\quad\) point: (3,10)
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