To analyze the path of the football described by the function \( h(x) = -130x(x - 60) \), we will work through the questions one by one.
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What is the y-intercept? Write your answer as an ordered pair.
The y-intercept occurs when \( x = 0 \).
\[ h(0) = -130(0)(0 - 60) = 0 \]
Therefore, the y-intercept is \( (0, 0) \).
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At what time is the football at the highest height?
The function \( h(x) \) is a quadratic equation which opens downwards ( since the coefficient of \( x^2 \) is negative), meaning it has a maximum point (the vertex). The x-coordinate of the vertex of a quadratic function of the form \( ax^2 + bx + c \) is given by the formula:
\[ x = -\frac{b}{2a} \]
For the function \( h(x) = -130x^2 + 7800x \):
- \( a = -130 \)
- \( b = 7800 \)
Thus,
\[ x = -\frac{7800}{2(-130)} = \frac{7800}{260} = 30 \]
Therefore, the football reaches its highest height at \( x = 30 \) yards.
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The ball will have traveled ____ yards before hitting the ground.
The football will hit the ground when \( h(x) = 0 \). We can solve for \( x \) when \( h(x) \) equals zero:
\[ h(x) = -130x(x - 60) = 0 \]
Setting each factor to zero gives:
\[ -130x = 0 \quad \Rightarrow \quad x = 0 \] \[ x - 60 = 0 \quad \Rightarrow \quad x = 60 \]
Therefore, the ball will hit the ground after traveling \( 60 \) yards.
To summarize the answers:
- The y-intercept is: \( (0, 0) \)
- The football reaches its highest height at: \( 30 \) yards
- The ball will have traveled: \( 60 \) yards before hitting the ground.
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