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Chapter 3: Problem 74

What is a quadratic function?

Short Answer

Expert verified
A quadratic function is a function that can be described by an equation of the form \( f(x) = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants and \( a \) is not zero. The graph of a quadratic function is a parabola.

Step by step solution

01

Definition of Quadratic Function

A quadratic function is a function that can be written in the standard form \( f(x) = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are real numbers and \( a \) is not equal to zero. The graph of a quadratic function is called a parabola.
02

Features of Quadratic Functions

In a quadratic function, the highest power of the variable (usually denoted as \( x \)) is 2. The standard form of the function allows us to calculate key features: the direction of the parabola (upward if \( a > 0 \), downward if \( a < 0 \)), the vertex (the maximum or minimum point), and the function's zeroes (the x-values where \( f(x) = 0 \)).
03

Examples of Quadratic Functions

Examples of quadratic functions include \( f(x) = 2x^2 + 3x + 1 \), where \( a = 2, b = 3, c = 1 \), and \( g(x) = -x^2 + 4x - 5 \), where \( a = -1, b = 4, c = -5 \). Both are valid quadratic functions, as their highest power of \( x \) is 2.

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Most popular questions from this chapter

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I must have made an error when graphing this parabola because its axis of symmetry is the \(y\) -axis.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I threw a baseball vertically upward and its path was a parabola.

Find the vertex for each parabola. Then determine a reasonable viewing rectangle on your graphing utility and use it to graph the quadratic function. $$y=-4 x^{2}+20 x+160$$

Use long division to rewrite the equation for \(g\) in the form $$\text {quotient }+\frac{\text {remainder}}{\text {divisor}}$$ Then use this form of the function's equation and transformations. $$g(x)=\frac{2 x-9}{x-4}$$

Describe how to find a parabola's vertex if its equation is expressed in standard form. Give an example.
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