Question

graph the function $f(x) = -5^x - 4$ on the axes below. you must plot the asymptote and any two points with integer coordinates.
asymptote:
vertical
horizontal

Explanation:

Step1: Analyze the function type

The function \( f(x) = -5^x - 4 \) is an exponential function. For exponential functions of the form \( y = a\cdot b^x + c \), the horizontal asymptote is determined by the constant term \( c \) as \( x \to \pm\infty \).

Step2: Find the horizontal asymptote

As \( x \to \infty \), \( 5^x \to \infty \), so \( -5^x \to -\infty \), and \( f(x)=-5^x - 4\to -\infty \). As \( x \to -\infty \), \( 5^x \to 0 \) (since any positive number to the power of a large negative number approaches 0), so \( -5^x\to 0 \), and \( f(x)=-5^x - 4\to 0 - 4=-4 \). So the horizontal asymptote is \( y = -4 \).

Step3: Find two points with integer coordinates

• For \( x = 0 \): \( f(0)=-5^0 - 4=-1 - 4=-5 \), so the point is \( (0, -5) \).
• For \( x = 1 \): \( f(1)=-5^1 - 4=-5 - 4=-9 \), so the point is \( (1, -9) \). (We could also use \( x=- 1 \): \( f(-1)=-5^{-1}-4=-\frac{1}{5}-4=-4.2 \), but this is not an integer coordinate, so we choose \( x = 0 \) and \( x = 1 \) for integer coordinates.)

Answer:

The asymptote is horizontal with equation \( y=-4 \). Two points with integer coordinates are \( (0, -5) \) and \( (1, -9) \) (the asymptote is horizontal, \( y = - 4 \)).