Question

graph the function $f(x) = -5^x$ 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 \( f(x) = -5^x \) is an exponential function. For exponential functions of the form \( a \cdot b^x + c \), the horizontal asymptote is determined by the limit as \( x \to \pm\infty \). In the case of \( y = -5^x \), as \( x \to \infty \), \( 5^x \to \infty \), so \( -5^x \to -\infty \); as \( x \to -\infty \), \( 5^x \to 0 \), so \( -5^x \to 0 \). Thus, the horizontal asymptote is \( y = 0 \) (the x - axis), which is a horizontal line.

Step2: Find two points with integer coordinates

• When \( x = 0 \):

Substitute \( x = 0 \) into the function \( f(x)=-5^{x} \). We know that any non - zero number to the power of 0 is 1, so \( f(0)=-5^{0}=- 1 \). So the point is \( (0, - 1) \).

• When \( x = 1 \):

Substitute \( x = 1 \) into the function \( f(x)=-5^{x} \). Then \( f(1)=-5^{1}=-5 \). So the point is \( (1, - 5) \). (We could also use \( x=- 1 \): \( f(-1)=-5^{-1}=-\frac{1}{5} \), but this does not have an integer y - coordinate. So \( (0, - 1) \) and \( (1, - 5) \) are better choices as they have integer coordinates.)

Answer:

Asymptote: Horizontal (the equation of the asymptote is \( y = 0 \)).
Two points: \( (0, - 1) \) and \( (1, - 5) \) (or other valid integer - coordinate points like \( (0, - 1) \) and \( (-1,-\frac{1}{5}) \) but the first two are with integer y - coordinates for \( x = 0,1 \)). To graph: Draw the horizontal line \( y = 0 \) (the x - axis) as the asymptote, then plot the points \( (0, - 1) \) and \( (1, - 5) \) and draw the curve of the exponential function \( f(x)=-5^{x} \) passing through these points and approaching the asymptote \( y = 0 \) as \( x\to-\infty \).