Answered

Find the best solutions to your questions at Westonci.ca, the premier Q&A platform with a community of knowledgeable experts. Get detailed answers to your questions from a community of experts dedicated to providing accurate information. Experience the ease of finding precise answers to your questions from a knowledgeable community of experts.

Graph the equation [tex][tex]$y = -x^2 + 2x + 3$[/tex][/tex] on the accompanying set of axes.

1. Plot 5 points, including the roots and the vertex.
2. Using the graph, determine the roots of the equation [tex]-x^2 + 2x + 3 = 0[/tex].

Click to plot points. Click points to delete them.

Sagot :

GinnyAnswer
To graph the equation [tex]\( y = -x^2 + 2x + 3 \)[/tex] and determine its roots and vertex, let's follow these steps:

1. Determine the roots of the equation [tex]\( -x^2 + 2x + 3 = 0 \)[/tex]:
The roots (solutions) of [tex]\( -x^2 + 2x + 3 = 0 \)[/tex] are:
- [tex]\( x = -1.0 \)[/tex]
- [tex]\( x = 3.0 \)[/tex]

Therefore, the points that you will plot for the roots are [tex]\((-1.0, 0)\)[/tex] and [tex]\((3.0, 0)\)[/tex].

2. Calculate the vertex of the parabola [tex]\( y = -x^2 + 2x + 3 \)[/tex]:
The vertex of the parabola, which is the maximum point because the parabola opens downwards, is given by the vertex coordinates:
- [tex]\( (1.0, 4.0) \)[/tex]

Therefore, the vertex point is [tex]\((1.0, 4.0)\)[/tex].

3. Calculate additional points for plotting:
Let's calculate the value of [tex]\(y\)[/tex] for a few selected [tex]\(x\)[/tex] values to get additional points:
- For [tex]\( x = -2 \)[/tex]:
[tex]\[ y = -(-2)^2 + 2(-2) + 3 = -4 - 4 + 3 = -5 \][/tex]
Point: [tex]\((-2, -5)\)[/tex]

- For [tex]\( x = -1 \)[/tex]:
[tex]\[ y = -(-1)^2 + 2(-1) + 3 = -1 - 2 + 3 = 0 \][/tex]
Point: [tex]\((-1, 0)\)[/tex]

- For [tex]\( x = 0 \)[/tex]:
[tex]\[ y = -(0)^2 + 2(0) + 3 = 3 \][/tex]
Point: [tex]\((0, 3)\)[/tex]

- For [tex]\( x = 1 \)[/tex]:
[tex]\[ y = -(1)^2 + 2(1) + 3 = -1 + 2 + 3 = 4 \][/tex]
Point: [tex]\((1, 4)\)[/tex]

- For [tex]\( x = 2 \)[/tex]:
[tex]\[ y = -(2)^2 + 2(2) + 3 = -4 + 4 + 3 = 3 \][/tex]
Point: [tex]\((2, 3)\)[/tex]

4. Plot these points on a set of axes:
- Roots: [tex]\( (-1.0, 0) \)[/tex] and [tex]\( (3.0, 0) \)[/tex]
- Vertex: [tex]\( (1.0, 4.0) \)[/tex]
- Additional points:
- Point: [tex]\( (-2, -5) \)[/tex]
- Point: [tex]\( (0, 3) \)[/tex]
- Point: [tex]\( (2, 3) \)[/tex]

5. Draw the parabola:
Use the plotted points to sketch the parabola. The parabola will be symmetrical around its vertex, which is at [tex]\( (1.0, 4.0) \)[/tex].

By plotting these points, you can clearly see the shape and position of the parabola on the graph. The graph will help you visualize that the roots are [tex]\(x = -1\)[/tex] and [tex]\(x = 3\)[/tex].
We hope this information was helpful. Feel free to return anytime for more answers to your questions and concerns. We hope this was helpful. Please come back whenever you need more information or answers to your queries. Thank you for trusting Westonci.ca. Don't forget to revisit us for more accurate and insightful answers.