Question\#2: Graph $y=2+2^{x-2}$ using $x=0,1,2,3,4$ (Count by 1 's up to 8 along the $y$ -axis) (5 Points)

Studdy Solution

STEP 1

What is this asking? We need to plot the graph of $y = 2 + 2^{x-2}$ for $x$ values from 0 to 4, using a scale of 1 on the x-axis and a scale of 1 up to 8 on the y-axis. Watch out! Don't forget the order of operations!
We need to calculate the exponent $x-2$ first, then apply $2$ to that power, and finally add $2$ .

STEP 2

1. Calculate y-values
2. Plot the points

STEP 3

Let's start by calculating the values of $y$ when $x = 0$ .
We substitute $x = 0$ into our equation: $y = 2 + 2^{0-2} = 2 + 2^{-2}$ .

STEP 4

Remember that a negative exponent means "one over"!
So, $2^{-2}$ is the same as $\frac{1}{2^2} = \frac{1}{4} = 0.25$ .
Therefore, when $x = 0$ , $y = 2 + 0.25 = \textbf{2.25}$ .

STEP 5

Now for $x = 1$ : $y = 2 + 2^{1-2} = 2 + 2^{-1} = 2 + \frac{1}{2} = \textbf{2.5}$ .
Awesome!

STEP 6

Next, $x = 2$ : $y = 2 + 2^{2-2} = 2 + 2^0$ .
Anything to the power of zero is one, so $2^0 = 1$ .
This gives us $y = 2 + 1 = \textbf{3}$ .

STEP 7

Let's keep going!
For $x = 3$ : $y = 2 + 2^{3-2} = 2 + 2^1 = 2 + 2 = \textbf{4}$ .

STEP 8

Finally, for $x = 4$ : $y = 2 + 2^{4-2} = 2 + 2^2 = 2 + 4 = \textbf{6}$ .
We've got all our $y$ values!

STEP 9

Now, let's plot these points on our graph.
Remember, we're counting by 1's up to 8 on the y-axis.

STEP 10

Our points are: $(0, \textbf{2.25})$ , $(1, \textbf{2.5})$ , $(2, \textbf{3})$ , $(3, \textbf{4})$ , and $(4, \textbf{6})$ .
Mark these points on your graph.

STEP 11

Once you have all the points marked, connect them with a smooth curve.
This curve represents the graph of $y = 2 + 2^{x-2}$ .

STEP 12

The graph of $y = 2 + 2^{x-2}$ is plotted with the points $(0, 2.25)$ , $(1, 2.5)$ , $(2, 3)$ , $(3, 4)$ , and $(4, 6)$ connected by a smooth curve.

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