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Answer :
Sure! Let's go through each part of the question step-by-step.
### a. Quadratic Equation in Factored Form
1. The streamer is launched 3 seconds after the fuse is lit and lands 8 seconds after it is lit. This means the x-intercepts (or roots) of the quadratic are 3 and 8.
2. The quadratic equation in factored form can be written using these x-intercepts:
[tex]\[
y = -(x - 3)(x - 8)
\][/tex]
The negative sign indicates that the parabola opens downwards.
### b. Vertex of the Function
1. The vertex of a quadratic function in standard form can be found using the midpoint of the x-intercepts, which gives the x-coordinate of the vertex.
2. Calculate the x-coordinate:
[tex]\[
\text{Vertex } x\text{-coordinate} = \frac{3 + 8}{2} = 5.5
\][/tex]
3. Find the y-coordinate by plugging the x-coordinate back into the equation:
[tex]\[
y = -((5.5 - 3)(5.5 - 8))
\][/tex]
[tex]\[
y = -((2.5)(-2.5)) = -(-6.25) = 6.25
\][/tex]
4. Thus, the vertex of the function is [tex]\((5.5, 6.25)\)[/tex].
### c. Multiplier for the Graph's Function
1. In cases where a graph's function does not match the vertex exactly or needs adjustments, sometimes the quadratic equation needs to be multiplied by a specific value to match the graph.
2. Unfortunately, without specific information from the graph such as how the vertex should match or any scaling information, we cannot specify a multiplier.
This explains each part of the problem. If there's any additional information provided by the graph, that might help determine the correct multiplier to align the function precisely with the graph, but based on the data given, we've reached the results detailed above.
### a. Quadratic Equation in Factored Form
1. The streamer is launched 3 seconds after the fuse is lit and lands 8 seconds after it is lit. This means the x-intercepts (or roots) of the quadratic are 3 and 8.
2. The quadratic equation in factored form can be written using these x-intercepts:
[tex]\[
y = -(x - 3)(x - 8)
\][/tex]
The negative sign indicates that the parabola opens downwards.
### b. Vertex of the Function
1. The vertex of a quadratic function in standard form can be found using the midpoint of the x-intercepts, which gives the x-coordinate of the vertex.
2. Calculate the x-coordinate:
[tex]\[
\text{Vertex } x\text{-coordinate} = \frac{3 + 8}{2} = 5.5
\][/tex]
3. Find the y-coordinate by plugging the x-coordinate back into the equation:
[tex]\[
y = -((5.5 - 3)(5.5 - 8))
\][/tex]
[tex]\[
y = -((2.5)(-2.5)) = -(-6.25) = 6.25
\][/tex]
4. Thus, the vertex of the function is [tex]\((5.5, 6.25)\)[/tex].
### c. Multiplier for the Graph's Function
1. In cases where a graph's function does not match the vertex exactly or needs adjustments, sometimes the quadratic equation needs to be multiplied by a specific value to match the graph.
2. Unfortunately, without specific information from the graph such as how the vertex should match or any scaling information, we cannot specify a multiplier.
This explains each part of the problem. If there's any additional information provided by the graph, that might help determine the correct multiplier to align the function precisely with the graph, but based on the data given, we've reached the results detailed above.
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