1. **Problem statement:** A soccer ball is kicked straight up, and its height is recorded every 0.25 seconds. We have the data points for time $t$ and height $h(t)$. 2. **Goal:** - (a) Create a scatter plot and draw a curve of best fit by hand (conceptual). - (b) Determine an equation for the curve of best fit. - (c) Use the equation to find the height at $t=0.4$ seconds. 3. **Step (a): Scatter plot and curve of best fit** - Plot the points $(0,0), (0.25,3.0), (0.5,4.8), (0.75,5.8), (1.0,6.2), (1.25,5.7), (1.5,4.6), (1.75,2.7), (2.0,0)$ on graph paper. - The points form a parabolic shape, rising to a peak at $t=1.0$ s and then falling. - The curve of best fit is a quadratic function of the form: $$h(t) = at^2 + bt + c$$ 4. **Step (b): Find the quadratic equation** - Use the vertex form of a parabola since the peak is at $t=1.0$, height $6.2$: $$h(t) = a(t - 1)^2 + k$$ where $k=6.2$ is the maximum height. - Since the parabola opens downward, $a < 0$. - Use a known point to find $a$, for example at $t=0$, $h=0$: $$0 = a(0 - 1)^2 + 6.2$$ $$0 = a(1)^2 + 6.2$$ $$a + 6.2 = 0$$ $$a = -6.2$$ - So the equation is: $$h(t) = -6.2(t - 1)^2 + 6.2$$ - Expand to standard form: $$h(t) = -6.2(t^2 - 2t + 1) + 6.2 = -6.2t^2 + 12.4t - 6.2 + 6.2 = -6.2t^2 + 12.4t$$ 5. **Step (c): Find height at $t=0.4$ seconds** - Substitute $t=0.4$ into the equation: $$h(0.4) = -6.2(0.4)^2 + 12.4(0.4)$$ Calculate: $$0.4^2 = 0.16$$ $$h(0.4) = -6.2 \times 0.16 + 12.4 \times 0.4 = -0.992 + 4.96 = 3.968$$ 6. **Final answer:** The height of the ball at 0.4 seconds is approximately $3.97$ meters.