Use technology to find points and then graph the function y, equals, 2, start superscript, x, minus, 4, end superscript, minus, 1, commay=2

Question

Use technology to find points and then graph the function y, equals, 2, start superscript, x, minus, 4, end superscript, minus, 1, commay=2 
x−4
 −1, following the instructions below.
Equation of asymptote:
y, equalsy=

Plot at least four points with integer coordinates that fit on the axes below

GPT-4o mini · Answer

To analyze the function $ y = 2^{(x - 4)} - 1 $ and find the points to plot, we can first identify the characteristics of this function and its asymptote. ### Step 1: Asymptote The function $ y = 2^{(x - 4)} - 1 $ approaches the line $ y = -1 $ as $ x $ approaches negative infinity. Therefore, the equation of the asymptote is: $$ y = -1 $$ ### Step 2: Finding Points To plot the function, we can calculate the $ y $ values for various integer $ x $ values. We'll choose a range of $ x $ values to see how the function behaves. Let's evaluate the function at $ x = 0, 1, 2, 3, 4, 5, 6, 7 $: - For $ x = 0 $: $$ y = 2^{(0 - 4)} - 1 = 2^{-4} - 1 = \frac{1}{16} - 1 = \frac{1 - 16}{16} = -\frac{15}{16} \approx -0.9375 $$ - For $ x = 1 $: $$ y = 2^{(1 - 4)} - 1 = 2^{-3} - 1 = \frac{1}{8} - 1 = \frac{1 - 8}{8} = -\frac{7}{8} = -0.875 $$ - For $ x = 2 $: $$ y = 2^{(2 - 4)} - 1 = 2^{-2} - 1 = \frac{1}{4} - 1 = \frac{1 - 4}{4} = -\frac{3}{4} = -0.75 $$ - For $ x = 3 $: $$ y = 2^{(3 - 4)} - 1 = 2^{-1} - 1 = \frac{1}{2} - 1 = \frac{1 - 2}{2} = -\frac{1}{2} = -0.5 $$ - For $ x = 4 $: $$ y = 2^{(4 - 4)} - 1 = 2^{0} - 1 = 1 - 1 = 0 $$ - For $ x = 5 $: $$ y = 2^{(5 - 4)} - 1 = 2^{1} - 1 = 2 - 1 = 1 $$ - For $ x = 6 $: $$ y = 2^{(6 - 4)} - 1 = 2^{2} - 1 = 4 - 1 = 3 $$ - For $ x = 7 $: $$ y = 2^{(7 - 4)} - 1 = 2^{3} - 1 = 8 - 1 = 7 $$ ### Step 3: Integer Points Let's focus on the integer coordinate points we found: - $ (4, 0) $ - $ (5, 1) $ - $ (6, 3) $ - $ (7, 7) $ ### Results Here are the integer coordinates we can plot: 1. $ (4, 0) $ 2. $ (5, 1) $ 3. $ (6, 3) $ 4. $ (7, 7) $ ### Graphing the Function On a coordinate plane, we can mark these points. The asymptote $ y = -1 $ will also be represented as a dashed horizontal line across the graph at $ y = -1 $. Here's a simple representation of the function: - The curve rises steeply after $ x = 4 $. - It approaches the asymptote (horizontal line at $ y = -1 $) for smaller values of $ x $. ### Final Summary - **Equation of Asymptote:** $ y = -1 $ - **Integer Points to Plot:** - $ (4, 0) $ - $ (5, 1) $ - $ (6, 3) $ - $ (7, 7) $ You can plot these points and sketch the curve for a complete view of the function behavior!