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Sagot :
### Part A: Finding the Vertex
To find the vertex of the quadratic function [tex]\( V(x) = 415x^2 - 4600x + 200000 \)[/tex], we need to use the vertex formula for a parabola, which is [tex]\( x = -\frac{b}{2a} \)[/tex], where [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are the coefficients from the standard quadratic equation [tex]\( ax^2 + bx + c \)[/tex].
1. Identify the coefficients from the given quadratic function:
- [tex]\( a = 415 \)[/tex]
- [tex]\( b = -4600 \)[/tex]
- [tex]\( c = 200000 \)[/tex]
2. Substitute [tex]\( a \)[/tex] and [tex]\( b \)[/tex] into the vertex formula:
[tex]\[ x = -\frac{-4600}{2 \cdot 415} = \frac{4600}{830} \][/tex]
3. Simplify the fraction:
[tex]\[ x = \frac{4600}{830} \approx 5.542168674698795 \][/tex]
So, the x-coordinate of the vertex is approximately [tex]\( 5.542168674698795 \)[/tex].
Next, we'll find the y-coordinate (V(x)) of the vertex by substituting the x-coordinate back into the quadratic function.
4. Substitute [tex]\( x \approx 5.542168674698795 \)[/tex] back into the quadratic function:
[tex]\[ V(5.542168674698795) = 415(5.542168674698795)^2 - 4600(5.542168674698795) + 200000 \][/tex]
5. Calculate the value:
[tex]\[ V(5.542168674698795) \approx 187253.01204819276 \][/tex]
So, the coordinates of the vertex are approximately [tex]\( (5.542168674698795, 187253.01204819276) \)[/tex].
### Part B: Interpreting the Vertex
The vertex [tex]\( (5.542168674698795, 187253.01204819276) \)[/tex] represents a key point in the context of the home's value over time.
- x-coordinate [tex]\( 5.542168674698795 \)[/tex]: This represents the number of years after 2020 when the home reaches its minimum value. Approximately 5.54 years after 2020 is around mid-2025.
- y-coordinate [tex]\( 187253.01204819276 \)[/tex]: This represents the minimum value of the home. The value of the home will be at its lowest point, which is approximately [tex]$187,253.01, during mid-2025. Thus, in terms of the value of the home, the vertex indicates that the home's value will decrease and reach its minimum value of about $[/tex]187,253.01 around the middle of the year 2025. After this point, if the model holds, the value of the home would begin to increase again.
To find the vertex of the quadratic function [tex]\( V(x) = 415x^2 - 4600x + 200000 \)[/tex], we need to use the vertex formula for a parabola, which is [tex]\( x = -\frac{b}{2a} \)[/tex], where [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are the coefficients from the standard quadratic equation [tex]\( ax^2 + bx + c \)[/tex].
1. Identify the coefficients from the given quadratic function:
- [tex]\( a = 415 \)[/tex]
- [tex]\( b = -4600 \)[/tex]
- [tex]\( c = 200000 \)[/tex]
2. Substitute [tex]\( a \)[/tex] and [tex]\( b \)[/tex] into the vertex formula:
[tex]\[ x = -\frac{-4600}{2 \cdot 415} = \frac{4600}{830} \][/tex]
3. Simplify the fraction:
[tex]\[ x = \frac{4600}{830} \approx 5.542168674698795 \][/tex]
So, the x-coordinate of the vertex is approximately [tex]\( 5.542168674698795 \)[/tex].
Next, we'll find the y-coordinate (V(x)) of the vertex by substituting the x-coordinate back into the quadratic function.
4. Substitute [tex]\( x \approx 5.542168674698795 \)[/tex] back into the quadratic function:
[tex]\[ V(5.542168674698795) = 415(5.542168674698795)^2 - 4600(5.542168674698795) + 200000 \][/tex]
5. Calculate the value:
[tex]\[ V(5.542168674698795) \approx 187253.01204819276 \][/tex]
So, the coordinates of the vertex are approximately [tex]\( (5.542168674698795, 187253.01204819276) \)[/tex].
### Part B: Interpreting the Vertex
The vertex [tex]\( (5.542168674698795, 187253.01204819276) \)[/tex] represents a key point in the context of the home's value over time.
- x-coordinate [tex]\( 5.542168674698795 \)[/tex]: This represents the number of years after 2020 when the home reaches its minimum value. Approximately 5.54 years after 2020 is around mid-2025.
- y-coordinate [tex]\( 187253.01204819276 \)[/tex]: This represents the minimum value of the home. The value of the home will be at its lowest point, which is approximately [tex]$187,253.01, during mid-2025. Thus, in terms of the value of the home, the vertex indicates that the home's value will decrease and reach its minimum value of about $[/tex]187,253.01 around the middle of the year 2025. After this point, if the model holds, the value of the home would begin to increase again.
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