graph the function $f(x) = -5^x$ on the axes below. you must plot the asymptote and any two points with…

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 ).