To analyze the soccer ball's trajectory represented by the function \( f(x) = -16x^2 + 25x \), we can find the x-intercepts (also known as the roots) by setting the function equal to zero:
\[ -16x^2 + 25x = 0 \]
Factoring out the common terms gives:
\[ x(25 - 16x) = 0 \]
Setting each factor to zero gives the x-intercepts:
- \( x = 0 \)
- \( 25 - 16x = 0 \) → \( 16x = 25 \) → \( x = \frac{25}{16} \) → \( x \approx 1.5625 \)
Thus, the x-intercepts are \( (0, 0) \) and \( \left(1.5625, 0\right) \).
Interpretation of the x-intercepts:
- First x-intercept (0, 0): This indicates the moment when the soccer ball is kicked from the ground (at time \( x = 0 \)).
- Second x-intercept \( \left(1.5625, 0\right) \): This indicates that the soccer ball will return to the ground approximately 1.5625 seconds after it has been kicked.
Graphing the function:
When you graph the function \( f(x) = -16x^2 + 25x \), it will be a downward-opening parabola with its vertex representing the maximum height of the ball. The vertex can be found using the formula \( x = -\frac{b}{2a} \):
\[ x = -\frac{25}{2 \cdot -16} = \frac{25}{32} \approx 0.78125 \]
Substituting this back into the function gives the maximum height:
\[ f\left(\frac{25}{32}\right) = -16\left(\frac{25}{32}\right)^2 + 25\left(\frac{25}{32}\right) \]
Calculating that gives the peak height.
Final Answer:
Given the analysis above, the x-intercept \( (1.563, 0) \) fits with our solution:
Response: \( \left(1.563, 0\right) \); The soccer ball will return to the ground 1.563 seconds after it has been kicked.