Question

8.1 quiz exponential functions

1. what is the value of h and k on the parent function: y = 6^(x - h)+k

a. h = 1, k = 3
b. h=-1, k=-3
c. h = 1, k=-3
d. h=-1, k = 3

1. what is the value of h and k and describe the effect on the parent function. y = 3^(x + 4)-2

a. h = 4 means 4 units to the left and k=-2 means 2 units down
b. h=-4 means 4 units to the left and k=-2 means 2 units down
c. h=-2 means 2 units up and k=-4 means 4 units down
d. h = 2 means 2 units down and k = 4 means 4 units up

1. find the value of a, the asymptote, determine if the graph has been compressed or stretched and if it has been reflected over the x or y - axis for: y=-\frac{1}{2}(3)^x-1

a. a = 1/2, asymptote y=-1, vertically stretched, reflected over the x - axis
b. a = 1/2, asymptote y = 1, vertically compressed, reflected over the y - axis
c. a=-1/2, asymptote y = 1, vertically compressed,
d. a=-1/2, asymptote y=-1, vertically compressed, reflected over the x - axis

1. determine the domain, range, y - intercept for the given graph

a. domain: all real numbers, range: y>4, y - intercept: (0,5)
b. domain: all real numbers, range: y>3, y - intercept: (0,7)
c. domain: all real numbers, range: y>3, y - intercept: (7,0)
d. domain: y>3, range: all real numbers, y - intercept: (0,7)

Explanation:

1. For the function \(y = 6^{x - 1}+3\)

Step1: Recall the form of an exponential - function \(y=a^{x - h}+k\)

The general form of an exponential function is \(y = a^{x - h}+k\), where \((h,k)\) is the horizontal and vertical shift of the parent - function \(y = a^{x}\). For the function \(y = 6^{x - 1}+3\), comparing with \(y=a^{x - h}+k\), we have \(h = 1\) and \(k = 3\).

Step1: Identify \(h\) and \(k\) in the form \(y=a^{x - h}+k\)

Rewrite \(y = 3^{x + 4}-2\) as \(y = 3^{x-(-4)}-2\). Here, \(h=-4\) and \(k = - 2\). The value of \(h=-4\) means a shift of 4 units to the left (since the shift is \(x - h\)), and \(k=-2\) means a shift of 2 units down.

Step1: Identify the value of \(a\)

In the exponential function \(y = a\cdot b^{x}+k\), for \(y=-\frac{1}{2}(3)^{x}-1\), \(a=-\frac{1}{2}\), \(b = 3\), and \(k=-1\).

Step2: Determine the asymptote

The horizontal asymptote of the function \(y=a\cdot b^{x}+k\) is \(y = k\). So, the asymptote is \(y=-1\).

Step3: Analyze the transformation

Since \(|a|=\frac{1}{2}<1\), the graph is vertically compressed. Since \(a<0\), the graph is reflected over the \(x\) - axis.

Answer:

A. \(h = 1,k = 3\)

2. For the function \(y = 3^{x + 4}-2\)