Problem 12 - Intermediate Algebra

Question

Fill in the blanks. a. The graph of \(g(x)=4^{x}+3\) is similar to the graph of \(f(x)=4^{\mathrm{r}},\) but it is translated 3 units____ b. The graph of \(g(x)=4^{x-3}\) is similar to the graph of \(f(x)=4^{x},\) but it is translated 3 units to the_____

Step-by-Step Solution

Answer

a. Upward; b. Right

Step 1: Understanding Vertical Translations

The graph of a function can be translated vertically by adding or subtracting a constant value to the entire function. For instance, if given \(g(x) = 4^x + 3\), it means the graph of the basic exponential function \(4^x\) is moved upwards by 3 units. This is because adding 3 increases the y-values of the function without altering the x-values.

Step 2: Analyzing the Given Function for Vertical Translation

In part a, the function \(g(x) = 4^x + 3\) involves adding 3 to \(4^x\), indicating a vertical translation. Therefore, the graph of \(g(x)\) is translated 3 units **upward** compared to \(f(x) = 4^x\).

Step 3: Understanding Horizontal Translations

The graph of a function can be translated horizontally by altering the input (x-value) with an additive or subtractive constant. In the function \(g(x) = 4^{x-3}\), this transformation affects the x-value, translating the graph horizontally without altering the y-values.

Step 4: Analyzing the Given Function for Horizontal Translation

For part b, the function \(g(x) = 4^{x-3}\) involves subtracting 3 from the input variable \(x\). This results in a rightward shift of the graph, translating it 3 units to the **right** compared to \(f(x) = 4^x\).

Key Concepts

Vertical Translation

Vertical translation refers to moving a graph up or down on the coordinate plane. If you add a constant to a function, this will shift the graph vertically. For example, consider the exponential function \( f(x) = 4^x \). If we modify this to \( g(x) = 4^x + 3 \), we are adding 3 to each y-value of the function. This results in the entire graph moving 3 units up.

• Adding a positive constant \(+c\) moves the graph upward.
• Adding a negative constant \(-c\) moves the graph downward.

The core element here is that the x-values stay unaffected by the translation. Only the y-values change, effectively raising or lowering the entire graph. Understanding this helps students visualize how alterations to the function can adjust its position without changing its shape.

Horizontal Translation

Horizontal translation occurs when you move a graph left or right by altering the x-values before applying the function. This change shifts the location of the graph on the x-axis while maintaining its vertical position. Taking the exponential function \( f(x) = 4^x \) and changing it to \( g(x) = 4^{x-3} \) moves the graph 3 units to the right.

• Subtracting a constant \(c\) from \(x\) shifts the graph to the right.
• Adding a constant \(c\) to \(x\) shifts it to the left.

This transformation only impacts the starting points and shape of the graph isn't affected. When you see an expression like \( 4^{x-c} \), focus on the alteration made to the input to predict the direction of the shift.

Exponential Function

An exponential function is a mathematical expression in which a constant base is raised to a variable power, typically written as \( f(x) = a^x \). These functions are known for their rapid rate of growth or decay and are represented graphically by a curve that rises or falls sharply. For example, \( f(x) = 4^x \) illustrates how as \(x\) increases, the value of \(4^x\) grows quickly compared to a linear function.
Key characteristics of exponential functions:

• They have a constant base and a variable exponent.
• Exponential growth is depicted as \(a > 1\), leading to an ascending curve.
• Exponential decay, when \(0 < a < 1\), produces a descending curve.

These functions model real-world phenomena such as population growth, radioactive decay, and compound interest, making them an essential element of mathematics education.