Question

graph the following function.
$y = 5 \cdot 2^{x}$
show your work here
draw

Explanation:

Step1: Identify base function form

This is an exponential function of the form $y = a \cdot b^x$, where $a=5$, $b=2$ (growth factor, $b>1$ so it's exponential growth).

Step2: Calculate key points

For $x=0$:

$y = 5 \cdot 2^0 = 5 \cdot 1 = 5$ → Point: $(0, 5)$

For $x=1$:

$y = 5 \cdot 2^1 = 5 \cdot 2 = 10$ → Point: $(1, 10)$

For $x=2$:

$y = 5 \cdot 2^2 = 5 \cdot 4 = 20$ (note: this is above the given graph's max y-value of 14, so we note it's outside the grid)

For $x=-1$:

$y = 5 \cdot 2^{-1} = 5 \cdot \frac{1}{2} = 2.5$ → Point: $(-1, 2.5)$

For $x=-2$:

$y = 5 \cdot 2^{-2} = 5 \cdot \frac{1}{4} = 1.25$ → Point: $(-2, 1.25)$

Step3: Identify asymptote

For $y = a \cdot b^x$, the horizontal asymptote is $y=0$ (as $x \to -\infty$, $y \to 0$).

Step4: Plot points and sketch

Plot the calculated points on the grid, then draw a smooth curve that approaches $y=0$ as $x$ decreases, passes through the plotted points, and increases rapidly as $x$ increases (extending beyond the grid for positive $x>1$).

Answer:

• Horizontal asymptote: $y=0$
• Plotted points: $(-2, 1.25)$, $(-1, 2.5)$, $(0, 5)$, $(1, 10)$
• The graph is an increasing exponential curve that approaches the x-axis (from above) as $x$ goes to negative infinity, and rises steeply as $x$ increases.