Question 5 Consider the function \( f(x)=-(5)^{x} \). Graph the function on a separate sheet of paper and identify the key features. Function type: \( \square \) \( y \)-Intercept: \( \square \) \( (0,-5) \) Domain: \( \square \) Range: \( \square \) \( \{y \mid y>0\} \) Increasing/Decreasing: always decreasing Positive/Negative: \( \square \) always negative Asymptote: \( \square \) \( y=0 \) End behavior As \( x \) increases, \( f(x) \) \( \square \) decreases and as \( x \) decreases, \( f(x) \) \( \square \) increases

Let's analyze the function \( f(x) = -5^{x} \) and identify its key features by filling in the blanks provided. ### Function Analysis 1. **Function Type:** - **Exponential Function** The given function is an exponential function because it has the form \( f(x) = a \cdot b^{x} \), where \( b = 5 \) and \( a = -1 \). 2. **\( y \)-Intercept:** - **\((0, -1)\)** To find the \( y \)-intercept, evaluate \( f(0) \): \[ f(0) = -5^{0} = -1 \] So, the \( y \)-intercept is at the point \( (0, -1) \). 3. **Domain:** - **All real numbers (\( \mathbb{R} \))** Exponential functions are defined for all real numbers \( x \). 4. **Range:** - **\(\{ y \mid y < 0 \}\)** Since the exponential function \( 5^{x} \) is always positive and it is multiplied by -1, \( f(x) \) is always negative. 5. **Increasing/Decreasing:** - **Always decreasing** The function decreases because the base \( 5 \) is greater than 1 and it's multiplied by a negative constant, causing the graph to decline as \( x \) increases. 6. **Positive/Negative:** - **Always negative** As established in the range, \( f(x) \) is always negative. 7. **Asymptote:** - **\( y = 0 \)** The horizontal asymptote for this exponential function is the \( x \)-axis, \( y = 0 \), which the graph approaches but never touches. 8. **End Behavior:** - **As \( x \) increases, \( f(x) \) decreases and as \( x \) decreases, \( f(x) \) increases.** - **As \( x \to \infty \):** \( f(x) = -5^{x} \to -\infty \) - **As \( x \to -\infty \):** \( f(x) = -5^{x} \to 0^{-} \) (approaches 0 from below) ### Summary of Key Features - **Function Type:** Exponential Function - **\( y \)-Intercept:** \( (0, -1) \) - **Domain:** All real numbers (\( \mathbb{R} \)) - **Range:** \( \{ y \mid y < 0 \} \) - **Increasing/Decreasing:** Always decreasing - **Positive/Negative:** Always negative - **Asymptote:** \( y = 0 \) (Horizontal Asymptote) - **End Behavior:** - As \( x \) increases, \( f(x) \) **decreases** (heads towards \( -\infty \)) - As \( x \) decreases, \( f(x) \) **increases** (approaches 0 from below) ### Graphing the Function When graphing \( f(x) = -5^{x} \): - **Start at the \( y \)-intercept (0, -1).** - **The graph decreases rapidly as \( x \) increases.** - **As \( x \) becomes more negative, the graph approaches the horizontal asymptote \( y = 0 \) but never touches it.** - **There are no \( x \)-intercepts since the function never crosses the \( x \)-axis.** Here's a rough sketch of the graph:  *(Please note: Replace the image link with an actual graph image if needed.)* ### Conclusion By filling in all the blanks correctly, we've identified that \( f(x) = -5^{x} \) is an always decreasing exponential function with a \( y \)-intercept at \( (0, -1) \), a domain of all real numbers, and a range of all negative real numbers. It has a horizontal asymptote at \( y = 0 \) and exhibits specific end behaviors as \( x \) approaches positive and negative infinity.