Answered

Welcome to Westonci.ca, where your questions are met with accurate answers from a community of experts and enthusiasts. Get immediate and reliable answers to your questions from a community of experienced professionals on our platform. Experience the convenience of finding accurate answers to your questions from knowledgeable experts on our platform.

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].
Thanks for using our platform. We're always here to provide accurate and up-to-date answers to all your queries. We hope you found what you were looking for. Feel free to revisit us for more answers and updated information. We're glad you visited Westonci.ca. Return anytime for updated answers from our knowledgeable team.