Let's start by completing a table of coordinates for the equation [tex]\( y = -2^x + 3 \)[/tex]. We will evaluate this function for different values of [tex]\( x \)[/tex] to find corresponding [tex]\( y \)[/tex]-values.
### Table of Coordinates
1. For [tex]\( x = -2 \)[/tex]:
[tex]\[
y = -2^{-2} + 3 = -\frac{1}{4} + 3 = 3 - 0.25 = 2.75
\][/tex]
2. For [tex]\( x = -1 \)[/tex]:
[tex]\[
y = -2^{-1} + 3 = -\frac{1}{2} + 3 = 3 - 0.5 = 2.5
\][/tex]
3. For [tex]\( x = 0 \)[/tex]:
[tex]\[
y = -2^0 + 3 = -1 + 3 = 2
\][/tex]
4. For [tex]\( x = 1 \)[/tex]:
[tex]\[
y = -2^1 + 3 = -2 + 3 = 1
\][/tex]
5. For [tex]\( x = 2 \)[/tex]:
[tex]\[
y = -2^2 + 3 = -4 + 3 = -1
\][/tex]
From these calculations, our table of coordinates is:
[tex]\[
\begin{array}{c|c}
x & y \\
\hline
-2 & 2.75 \\
-1 & 2.5 \\
0 & 2 \\
1 & 1 \\
2 & -1 \\
\end{array}
\][/tex]
### Plotting the Graph
Based on the coordinates calculated, we can plot these points on a graph:
- Point 1: [tex]\((-2, 2.75)\)[/tex]
- Point 2: [tex]\((-1, 2.5)\)[/tex]
- Point 3: [tex]\((0, 2)\)[/tex]
- Point 4: [tex]\((1, 1)\)[/tex]
- Point 5: [tex]\((2, -1)\)[/tex]
### Shape of [tex]\( y = -2^x + 3 \)[/tex]
To draw the graph, plot the points on the Cartesian plane and then draw a smooth curve through them to represent the equation [tex]\( y = -2^x + 3 \)[/tex]:
1. Start at point [tex]\((-2, 2.75)\)[/tex] and move to [tex]\((-1, 2.5)\)[/tex].
2. Continue from [tex]\((-1, 2.5)\)[/tex] to [tex]\((0, 2)\)[/tex].
3. Move from [tex]\((0, 2)\)[/tex] to [tex]\((1, 1)\)[/tex].
4. Finally, draw from [tex]\((1, 1)\)[/tex] to [tex]\((2, -1)\)[/tex].
The curve will showcase a rapid decrease as [tex]\( x \)[/tex] increases because the function [tex]\( -2^x \)[/tex] grows exponentially more negative. The addition of [tex]\( 3 \)[/tex] shifts the entire plot up by 3 units on the [tex]\( y \)[/tex]-axis.
### Interpretation of the Equation
The graph of the function [tex]\(-2^x + 3\)[/tex] exhibits the following:
- A downward slope from left to right, indicating the negative exponential component [tex]\(-2^x\)[/tex].
- The curve intersects the [tex]\( y \)[/tex]-axis at [tex]\( y = 2 \)[/tex], since when [tex]\( x = 0 \)[/tex], [tex]\( y = -2^0 + 3 = 2 \)[/tex].
- The curve passes through the points calculated from the table of coordinates, and as [tex]\( x \)[/tex] increases, [tex]\( y \)[/tex] becomes increasingly negative due to the [tex]\( -2^x \)[/tex] term.
Ensure when plotting this, the graph reflects the steep nature of exponential functions, becoming sharply negative as [tex]\( x \)[/tex] increases.
