**Quadratic Function and Discriminant Analysis**Let $ f(x) = ax^2 + bx + c $, and let $ D = b^2 - 4ac $.(a) If $ D > 0 $, then the number of x-intercepts for the graph of $ f $ is ______.(b) If $ D = 0 $, then the number of x-intercepts for the graph of $ f $ is 0.(c) If $ D < 0 $, then the number of x-intercepts for the graph of $ f $ is ______.**Explanation:**- **Discriminant ($ D $)**: The discriminant of a quadratic equation $ ax^2 + bx + c = 0 $ is given by $ D = b^2 - 4ac $.- **X-intercepts**: The x-intercepts of a quadratic function are the solutions to the equation $ ax^2 + bx + c = 0 $.**Analysis:**1. **$ D > 0 $**: Indicates two distinct real x-intercepts (roots).2. **$ D = 0 $**: Indicates one real x-intercept (repeated root).3. **$ D < 0 $**: Indicates no real x-intercepts (roots are complex).In part (b), the text incorrectly suggests there are 0 x-intercepts when $ D = 0 $. It should state that there is 1 x-intercept.

Answered: Let f(x) = ax² + bx + c, and let D =…

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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**Quadratic Function and Discriminant Analysis**

Let $ f(x) = ax^2 + bx + c $, and let $ D = b^2 - 4ac $.

(a) If $ D > 0 $, then the number of x-intercepts for the graph of $ f $ is ______.

(b) If $ D = 0 $, then the number of x-intercepts for the graph of $ f $ is 0.

(c) If $ D < 0 $, then the number of x-intercepts for the graph of $ f $ is ______.

**Explanation:**

- **Discriminant ($ D $)**: The discriminant of a quadratic equation $ ax^2 + bx + c = 0 $ is given by $ D = b^2 - 4ac $.
- **X-intercepts**: The x-intercepts of a quadratic function are the solutions to the equation $ ax^2 + bx + c = 0 $.

**Analysis:**

1. **$ D > 0 $**: Indicates two distinct real x-intercepts (roots).
2. **$ D = 0 $**: Indicates one real x-intercept (repeated root).
3. **$ D < 0 $**: Indicates no real x-intercepts (roots are complex).

In part (b), the text incorrectly suggests there are 0 x-intercepts when $ D = 0 $. It should state that there is 1 x-intercept.

Expert Solution

Step 1

The number of x-intercepts of the function $f (x) = a x^{2} + b x + c$ , is equal to the number of distinct solutions of the equation $f (x) = 0$ .

Use the quadratic formula to find the solution of the equation $f (x) = 0$ .

$\begin{array}{rcl} f (x) & = & 0 \\ a x^{2} + b x + c & = & 0 \\ x & = & \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a} \\ = & \frac{- b \pm \sqrt{D}}{2 a} \end{array}$

Therefore, the x-intercepts of the function $f (x) = a x^{2} + b x + c$ is $(\frac{- b \pm \sqrt{D}}{2 a}, 0)$ .

Step 2

If $D > 0$ , then x has two distinct real values, that is $x = \frac{- b + \sqrt{D}}{2 a}$ and $x = \frac{- b - \sqrt{D}}{2 a}$ .

Hence, the x-intercepts of the function $f (x) = a x^{2} + b x + c$ are $(\frac{- b + \sqrt{D}}{2 a}, 0)$ and $(\frac{- b - \sqrt{D}}{2 a}, 0)$ .

Therefore, the number of x-intercepts of the graph $f (x) = a x^{2} + b x + c$ is 2.

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