Part 4 - Investigate the relationship between yy = aaxx aaaaaa yy = aa−xx wwwwwwwwww aa > 00

(3 marks)

Choose graphs of your own and investigate the effect of x and –x.

a) What is the y-intercept of each graph?

b) What is the horizontal asymptote of each graph?

c) What transformation moves yy = 2xx tt yy = 2−xx?

Let's explore the functions \( y = a^x \) and \( y = a^{-x} \), where \( a > 0 \). These are exponential functions, but the presence of \( x \) and \( -x \) will have different effects on the graph.

### 1. **Graphing \( y = a^x \) and \( y = a^{-x} \)**

Let's choose a few values of \( a \) to graph these functions and investigate the effects:
- \( y = 2^x \) and \( y = 2^{-x} \)
- \( y = 3^x \) and \( y = 3^{-x} \)
- \( y = \frac{1}{2}^x \) and \( y = \frac{1}{2}^{-x} \)

### Understanding the Graphs:

1. **\( y = a^x \)**: This graph is an exponential curve that increases rapidly as \( x \) increases (for \( a > 1 \)) and decreases slowly as \( x \) decreases.
2. **\( y = a^{-x} \)**: This graph is a reflection of \( y = a^x \) across the \( y \)-axis. As \( x \) increases, the graph decreases, and as \( x \) decreases, the graph increases.

### 2. **Answering the Questions**

#### a) What is the y-intercept of each graph?

The y-intercept is the point where the graph crosses the \( y \)-axis, which occurs when \( x = 0 \).

- For \( y = a^x \) and \( y = a^{-x} \), when \( x = 0 \):
  - \( y = a^0 = 1 \)
  - Therefore, **the y-intercept of each graph is \( y = 1 \)**, regardless of the value of \( a \).

#### b) What is the horizontal asymptote of each graph?

The horizontal asymptote is the value that the graph approaches as \( x \) approaches infinity or negative infinity.

- **For \( y = a^x \)** (with \( a > 1 \)):
  - As \( x \to \infty \), \( y \to \infty \).
  - As \( x \to -\infty \), \( y \to 0 \).
  - **Horizontal asymptote**: \( y = 0 \).

- **For \( y = a^{-x} \)**:
  - As \( x \to \infty \), \( y \to 0 \).
  - As \( x \to -\infty \), \( y \to \infty \).
  - **Horizontal asymptote**: \( y = 0 \).

So, in both cases, the horizontal asymptote is \( y = 0 \).

#### c) What transformation moves \( y = 2^x \) to \( y = 2^{-x} \)?

The transformation that changes \( y = 2^x \) to \( y = 2^{-x} \) is a **reflection across the y-axis**. 

- \( y = 2^{-x} \) is the reflection of \( y = 2^x \) over the y-axis.

### Questions for Further Exploration:

1. How does the graph of \( y = 3^x \) compare with \( y = 3^{-x} \)?
2. What happens to the graphs when \( a < 1 \), such as \( y = \frac{1}{2}^x \) and \( y = \frac{1}{2}^{-x} \)?
3. How does a vertical shift affect the graphs \( y = a^x \) and \( y = a^{-x} \)?
4. What is the effect of adding a constant to \( x \) or \( -x \) in the exponent?
5. How does the growth rate of \( y = e^x \) compare with \( y = e^{-x} \)?

### Tip:

When dealing with exponential functions, remember that reflecting across the y-axis flips the graph, while reflecting across the x-axis flips the output values. Always check the direction and magnitude of the reflection to understand the transformation.

Explore the effects of x and -x in exponential functions. Investigate y-intercepts, horizontal asymptotes, and transformations between functions y = a^x and y = a^{-x}. Learn how these graphs change with varying values of a. Suitable for advanced high school students studying exponential functions.

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