Problem 62 - Algebra 2

Question

REASONING Identify each function as linear, quadratic, or exponential. $$\begin{array}{llll}{\text { a. } y=3 x^{2}} & {\text { b. } y=4(3)^{x}} & {\text { c. } y=2 x+4} & {\text { d. } y=4(0.2)^{x}+1}\end{array}$$

Step-by-Step Solution

Answer

a is quadratic, b is exponential, c is linear, d is exponential.

Step 1: Analyze the First Function

The first function given is \( y = 3x^2 \). This function contains a variable \( x \) that is raised to the power of 2. Functions of this form are known as quadratic functions. A key feature of quadratic functions is the squared term \( x^2 \).

Step 2: Analyze the Second Function

The second function is \( y = 4(3)^x \). Notice here that the variable \( x \) is in the exponent. Functions where the variable is the exponent are categorized as exponential functions. Therefore, \( y = 4(3)^x \) is an exponential function.

Step 3: Analyze the Third Function

The third function is \( y = 2x + 4 \). In this expression, \( x \) is raised to the power of 1, and the function forms a straight line when graphed. A linear function has the form \( y = mx + b \), where \( m \) and \( b \) are constants. Thus, \( y = 2x + 4 \) is a linear function.

Step 4: Analyze the Fourth Function

The fourth function is \( y = 4(0.2)^x + 1 \). Similar to the second function, here \( x \) is in the exponent, making it an exponential function, as exponential functions have the form \( y = a(b)^x \) where \( a \) and \( b \) are constants.

Key Concepts

Linear Functions

Linear functions are one of the simplest types of functions, and they're essential building blocks in algebra. In a linear function, the highest power of the variable is 1. This means that the function has a constant rate of change, leading to a straight line when graphed. The general form of a linear function is \[ y = mx + b \]where

• \( m \) represents the slope or gradient of the line, which dictates its steepness.
• \( b \) is the y-intercept, the point where the line crosses the y-axis.

This function is useful for modeling relationships where one quantity changes at a constant rate with respect to another. For example, if you were graphing the function \( y = 2x + 4 \), you would start at the point (0, 4) on the y-axis and use the slope \( m = 2 \) to determine the direction and inclination of the line. Linear functions are ideal for simple calculations like budget projections, distance over time, and any other scenarios involving stable, consistent increases or decreases.

Quadratic Functions

Quadratic functions introduce complexity by having a variable that is raised to the power of 2. This results in a parabolic graph, which could open upwards or downwards. The most common form of a quadratic function is:\[ y = ax^2 + bx + c \] where:

• \( a \), \( b \), and \( c \) are constants.
• If \( a > 0 \), the parabola opens upwards, like a smile. If \( a < 0 \), it opens downwards, like a frown.
• The vertex of the parabola is the highest or lowest point, depending on the orientation of the curve.
Quadratic functions can model phenomena with accelerating or decelerating change. A classic example is the equation \( y = 3x^2 \) which forms a parabola opening upwards. Such quadratic equations are commonly used to describe the motion of objects under uniform acceleration, such as a projectile in physics. Recognizing the vertex, axis of symmetry, and intercepts are crucial to understanding the shape and key attributes of a parabola.

Exponential Functions

Exponential functions are defined by their characteristic feature where the variable appears in the exponent. These functions can either grow or decay very rapidly, depending on the value of the base. The general form of an exponential function is:\[ y = a(b)^x \]where:

• \( a \) is a constant representing the initial value or y-intercept.
• \( b \) is the base of the exponential and determines growth or decay.
• If \( b > 1 \), the function models exponential growth. If \( 0 < b < 1 \), it models exponential decay.

Exponential functions are immensely applicable in real-world scenarios, such as population growth, radioactive decay, interest compounding, and more. For example, in the function \( y = 4(3)^x \), because the base \( 3 \) is greater than 1, the function will show exponential growth. Conversely, in \( y = 4(0.2)^x + 1 \), the base of \( 0.2 \) signifies that the function is experiencing exponential decay. Understanding these functions helps in predicting and analyzing any processes that involve rapid changes over time.