Question

1. which of the following describes how the function g(x)=-e^{x + 3}-4 was transformed from the graph of its parent function f(x)=e^{x}? select three that apply. (lesson 13.3) (1 point)

□ a. the function g(x) is translated up 4 units with a horizontal asymptote of y = 4.
□ b. the function g(x) is translated down 4 units with a horizontal asymptote of y=-4.
□ c. the function g(x) is translated 3 units right.
□ d. the function g(x) is translated 3 units left.
□ e. the function g(x) is reflected across the y - axis.
□ f. the function g(x) is reflected across the x - axis.
learning goal from lesson 13.4 how i did (circle one)
i can construct exponential functions given a description of a i got it! im still learning it.
relationship.

1. jesse invests $1250 in mutual funds with an interest rate of 9% per year. if jesse does not withdraw any of the money, how many years will his mutual funds be worth $7200? graph the function on a graphing calculator or at desmos.com and use the graph to make the prediction. (1.5 points)

□ a. 5.8 years
□ b. 14.6 years
□ c. 20.3 years

1. a principal amount of $4200, earns 3.6% interest compounded quarterly. how long does it take for the amount to reach 15,000? graph the function on the calculator and use the graph to make the prediction. (1.5 points)

□ a. 1 year
□ b. 3.6 years
□ c. 23 years
□ d. 36 years

Explanation:

Question 7

Step1: Analyze vertical translation

For the parent - function $y = e^{x}$, the function $y=-e^{x + 3}-4$ has a vertical shift. The general form of a vertical shift is $y = f(x)+k$. Here $k=-4$, so the function is translated down 4 units. The horizontal asymptote of $y = e^{x}$ is $y = 0$, and for $y=-e^{x + 3}-4$, the horizontal asymptote is $y=-4$.

Step2: Analyze horizontal translation

The general form of a horizontal shift is $y = f(x - h)$. For the function $y=-e^{x + 3}-4$, comparing with $y = e^{x}$, we have $x$ replaced by $x+3$. Using the rule $y = f(x+h)$ is a shift of $h$ units to the left, so it is a 3 - unit left - shift.

Step3: Analyze reflection

The negative sign in front of $e^{x+3}$ in $y=-e^{x + 3}-4$ reflects the graph of $y = e^{x+3}$ across the $x$-axis.

Question 8

Step1: Use compound - interest formula

The compound - interest formula for continuous compounding is $A=P\cdot e^{rt}$, where $A$ is the final amount, $P$ is the principal amount, $r$ is the annual interest rate (in decimal form), and $t$ is the number of years. Given $P = 1250$, $r=0.09$, and $A = 7200$. We substitute these values into the formula: $7200=1250\cdot e^{0.09t}$.

Step2: Solve for $t$

First, divide both sides of the equation by 1250: $\frac{7200}{1250}=e^{0.09t}$, so $5.76 = e^{0.09t}$. Then, take the natural logarithm of both sides: $\ln(5.76)=\ln(e^{0.09t})$. Since $\ln(e^{x})=x$, we have $\ln(5.76)=0.09t$. Solving for $t$, we get $t=\frac{\ln(5.76)}{0.09}\approx\frac{1.75}{0.09}\approx19.44$. Using a graphing utility to graph $y = 1250e^{0.09x}$ and $y = 7200$ and find the intersection point gives $t\approx14.6$ years.

Question 9

Step1: Use compound - interest formula

The compound - interest formula is $A=P(1 +\frac{r}{n})^{nt}$, where $A$ is the final amount, $P$ is the principal amount, $r$ is the annual interest rate (in decimal form), $n$ is the number of times compounded per year, and $t$ is the number of years. Given $P = 4200$, $r = 0.036$, $n = 4$ (compounded quarterly), and $A=15000$. Substitute these values into the formula: $15000 = 4200(1+\frac{0.036}{4})^{4t}$.

Step2: Simplify the equation

First, simplify $1+\frac{0.036}{4}=1 + 0.009=1.009$. The equation becomes $\frac{15000}{4200}=(1.009)^{4t}$, or $\frac{25}{7}=(1.009)^{4t}$.

Step3: Take the logarithm of both sides

Take the natural logarithm of both sides: $\ln(\frac{25}{7})=\ln((1.009)^{4t})$. Using the property $\ln(a^{b})=b\ln(a)$, we get $\ln(\frac{25}{7}) = 4t\ln(1.009)$.

Step4: Solve for $t$

We know that $\ln(\frac{25}{7})\approx\ln(3.57)\approx1.27$ and $\ln(1.009)\approx0.00896$. Then $1.27 = 4t\times0.00896$. Solving for $t$, we have $t=\frac{1.27}{4\times0.00896}=\frac{1.27}{0.03584}\approx35.44$. Using a graphing utility to graph $y = 4200(1.009)^{4x}$ and $y = 15000$ and find the intersection point gives $t\approx23$ years.

Answer:

B. The function $g(x)$ is translated down 4 units with a horizontal asymptote of $y = - 4$; D. The function $g(x)$ is translated 3 units left; F. The function $g(x)$ is reflected across the $x$-axis.