From personal advice to professional guidance, IDNLearn.com has the answers you seek. Get the information you need from our experts, who provide reliable and detailed answers to all your questions.
Sagot :
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.
### 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.
Thank you for participating in our discussion. We value every contribution. Keep sharing knowledge and helping others find the answers they need. Let's create a dynamic and informative learning environment together. Thank you for choosing IDNLearn.com for your queries. We’re committed to providing accurate answers, so visit us again soon.
