Quadratic Functions Overview

Which is a quadratic function?

What are the values of the coefficients and constant in the function f(x) = x^2 - 5x + 6?

Estimate f(-3) for the given graph of f(x).

Evaluate the function f(x) at an input of 0.

What is the leading coefficient of the function f(x) = -2x^2 + 5x - 4?

Which table represents a quadratic function?

Find f(-2) for the function f(x) = 3x^2 - 2x + 7.

What is the rate of change for the interval between -6 and -3 on the x-axis?

Evaluate the function f(x) = -2x^2 - 3x + 5 for the input value -3.

What is the rate of change for the interval between 2 and 6 on the x-axis?

Which graph has a rate of change equal to 1/3 in the interval between 0 and 3 on the x-axis?

Which function increases at a faster rate on 0 to infinity, f(x) = x^2 or g(x) = 2^x? Explain your reasoning.

Quadratic Functions Overview

A quadratic function is in the form f(x) = ax² + bx + c, where a, b, and c are constants.
Valid examples of quadratic functions include f(x) = 5x² - 4x + 5, while functions like f(x) = 2x + x + 3 are linear.

Coefficients and Constants

For the function f(x) = x² - 5x + 6:
- a = 1 (coefficient of x²)
- b = -5 (coefficient of x)
- c = 6 (constant term)

Graph Interpretation

Estimating values such as f(-3) from a graph can yield results like 12.5.
Evaluating f(0) typically might give a value like 4 based on the function.

Leading Coefficient

The leading coefficient indicates how the function opens and its direction. For f(x) = -2x² + 5x - 4, the leading coefficient is -2, indicating it opens downwards.

Identifying Quadratic Functions

A table of values may help identify quadratic relationships by observing the changes in outputs. Look for consistent second differences.

Functional Evaluation

Finding values like f(-2) for the quadratic function f(x) = 3x² - 2x + 7 could render outcomes of 23.
Evaluating f(-3) for f(x) = -2x² - 3x + 5 can yield results like -4.

Rate of Change

The rate of change for segments on a graph can be calculated between intervals. For example, the rate of change between -6 and -3 is -2, while between 2 and 6, it can be 3.

Comparison of Growth Rates

Comparing functions f(x) = x² and g(x) = 2x reveals that g(x) increases at a faster, exponential rate than the quadratic function due to a constant multiplicative rate.

Graph Characteristics

Identifying which graph exhibits a specific rate of change, such as 1/3 between defined intervals, aids in understanding function behavior.

Explore the fundamentals of quadratic functions, including their standard form and key components like coefficients and constants. This quiz covers how to evaluate these functions and interpret their graphs, as well as identifying quadratic relationships through tables of values.