To analyze the function \( f(x) = -16x^2 + 25x \), we can follow these steps:
1. Identify the x-intercepts
The x-intercepts of the function occur when \( f(x) = 0 \). Therefore, we set the equation to zero and solve for \( x \):
\[ -16x^2 + 25x = 0 \]
Factoring out \( x \):
\[ x(-16x + 25) = 0 \]
Setting each factor equal to zero gives us the solutions:
- \( x = 0 \)
- \(-16x + 25 = 0 \rightarrow x = \frac{25}{16} = 1.5625 \)
Thus, the x-intercepts are \( x = 0 \) and \( x = 1.5625 \).
2. Interpret the x-intercepts
The function \( f(x) \) represents the height of the soccer ball (in feet) \( x \) seconds after being kicked.
- The first x-intercept at \( x = 0 \) represents the moment the ball is kicked. At this point, the height of the ball is \( f(0) = 0 \) feet (it is on the ground).
- The second x-intercept at \( x = 1.5625 \) seconds indicates that the ball reaches the ground again after being in the air for approximately 1.56 seconds.
3. Graph the function
To graph \( f(x) = -16x^2 + 25x \):
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Identify the vertex: The vertex of a quadratic function in the form \( ax^2 + bx + c \) is given by \( x = -\frac{b}{2a} \). Here, \( a = -16 \) and \( b = 25 \):
\[ x = -\frac{25}{2 \cdot -16} = \frac{25}{32} \approx 0.78125 \]
We can find the maximum height by substituting this \( x \) value back into the function:
\[ f\left(\frac{25}{32}\right) = -16\left(\frac{25}{32}\right)^2 + 25\left(\frac{25}{32}\right) \]
Calculating each term:
\[ f\left(\frac{25}{32}\right) = -16 \cdot \frac{625}{1024} + \frac{625}{32} \]
Finding a common denominator (1024):
\[ = -\frac{10000}{1024} + \frac{19688}{1024} = \frac{9688}{1024} \approx 9.46 \text{ feet} \]
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Plotting Points: In addition to the x-intercepts and the vertex, you can choose additional points to confirm the shape of the graph. You could select \( x = 0.5 \), \( x = 1 \), and \( x = 2 \).
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Shape of the Graph: The graph is a downward-opening parabola because the coefficient of \( x^2 \) is negative.
4. Summary
The graph of the function \( f(x) = -16x^2 + 25x \) describes the height of the soccer ball over time. The ball is kicked from the ground (at \( x = 0 \)) and reaches a maximum height (approximately 9.46 feet) before falling back down to the ground at approximately \( x = 1.56 \) seconds.