1.3 Rates of Change in Linear and Quadratic Functions - Top Study Guide

Rates of Change in Linear and Quadratic Functions

The rate of change in linear and quadratic functions helps us understand how the output of these functions changes in relation to changes in their input values. Linear functions have a constant rate of change, which is the slope of the line. Quadratic functions have a variable rate of change, showing acceleration or deceleration in the change of the output values.

Examples

Example 1: Linear Function

Function: y = 2x + 3

This linear function has a constant rate of change (slope) of 2. This means for every one unit increase in x, y increases by 2 units.

Example 2: Linear Function

Function: y = -4x + 1

The rate of change (slope) for this linear function is -4. For every one unit increase in x, y decreases by 4 units, indicating a downward slope.

Example 3: Quadratic Function

Function: y = x^2

The rate of change for this quadratic function varies with x. As x increases or decreases from 0, the rate of change increases, indicating the parabola's widening.

Example 4: Quadratic Function

Function: y = -x^2 + 4

This quadratic function opens downward (because of the negative coefficient of x^2), and its rate of change shows deceleration as x moves away from the vertex at (0, 4).

Example 5: Linear Function in a Real-World Context

Equation: y = 50x

If y represents the total cost of buying x apples at a constant price of $50 per apple, the rate of change, or slope, is 50. This indicates that the cost (output) increases by $50 for every additional apple (input) purchased.

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Corrective Assignments

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AP Learning Objectives:
1.3.A Determine the average rates of change for linear and quadratic sequences and functions.
1.3.B Determine the change of average rates of change for linear and quadratic functions.

Frequently Asked Questions: Rate of Change in Different Equations

What is the rate of change in a linear equation? How do you find it?

In a linear equation (like $y = mx + b$ or $Ax + By = C$), the rate of change is **constant**. It represents how much the dependent variable (usually $y$) changes for every one-unit increase in the independent variable (usually $x$).

In the standard slope-intercept form $y = mx + b$, the rate of change is simply the **slope of the line**, which is represented by the variable $m$.

If the equation is not in slope-intercept form, you can find the rate of change by:

Rearranging the equation into $y = mx + b$ form to identify $m$.
Finding the coordinates of any two points $(x_1, y_1)$ and $(x_2, y_2)$ that satisfy the equation and using the slope formula: $ \frac{y_2 - y_1}{x_2 - x_1} $.

For a linear equation, the rate of change is the same between any two points.

What is the rate of change in a quadratic equation? How do you find it?

In a quadratic equation (like $y = ax^2 + bx + c$), the rate of change is **not constant**. The graph of a quadratic equation is a parabola, and its slope (rate of change) is continuously changing.

To find the rate of change in a quadratic equation, you usually need to specify whether you want the:

Average Rate of Change over a specific interval $[a, b]$. You find this using the formula $ \frac{f(b) - f(a)}{b - a} $, just like with any function. This gives you the average slope between the two points.
Instantaneous Rate of Change at a specific point $x=a$. This requires calculus and is found by taking the derivative of the quadratic function, $f'(x)$, and evaluating it at $x=a$. The derivative of $y = ax^2 + bx + c$ is $y' = 2ax + b$. Evaluating this at a specific point gives the instantaneous rate of change at that point.

Without calculus, you can only find the *average* rate of change over an interval for a quadratic function.

What does the rate of change tell you about the graph of an equation?

The rate of change represents the **slope** of the graph at a particular point or over an interval.

A **positive** rate of change indicates the graph is increasing (sloping upwards).
A **negative** rate of change indicates the graph is decreasing (sloping downwards).
A rate of change of **zero** indicates the graph is momentarily flat (horizontal).
A **larger absolute value** of the rate of change indicates a steeper slope (faster change).

For linear equations, the constant rate of change means the graph is a single straight line. For quadratic equations, the changing rate of change means the slope varies, creating a curved shape (parabola).