To identify which graph represents the equation of the projectile's maximum height modeled by the equation \( h = -4.9t^2 + 15t \), we need to analyze the equation and its characteristics:
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Shape of the Graph: The equation is a quadratic function where the coefficient of the \( t^2 \) term is negative (-4.9). This indicates that the graph opens downward, representing a parabolic trajectory.
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Intercepts:
- When \( t = 0 \), \( h = -4.9(0)^2 + 15(0) = 0 \). So the graph passes through the point (0, 0).
- To find when \( h = 0 \), we can solve: \[ -4.9t^2 + 15t = 0 \ t(15 - 4.9t) = 0 \] This gives \( t = 0 \) or \( t = \frac{15}{4.9} \approx 3.06 \). Therefore, the other point where the graph crosses the t-axis is approximately (3, 0).
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Maximum Height: The vertex of this downward-opening parabola will give us the maximum height. The vertex \( t \) value can be calculated using \( t = -\frac{b}{2a} = -\frac{15}{2(-4.9)} \approx 1.53 \). We can substitute this value back into the height equation to find the maximum height, but it's generally sufficient to note that the curve reaches its peak between \( (0,0) \) and \( (3,0) \).
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Limits on Height: Since the equation gives \( h \leq -4.9t^2 + 15t \), the graph should also show shading below the curve.
Based on this analysis:
- The graph must open downward.
- It must pass through the points (0, 0) and approximately (3, 0).
- The shading must be below the curve.
Looking at the options you've provided:
- The first option matches these attributes, as it is a solid curve that opens downward and correctly passes through the important points while having shading below it.
Thus, the correct graph representing the projectile's height equation is the first option:
The graph of a solid curve that opens downward and goes through (0, 0) and (3, 0) with shading below the curve.
1 answer