Question

projectile selena kept track of the height of a ball after throwing it, where x is the time in seconds and f(x) is the height in feet. x | f(x) 0 | 6 0.25 | 11 0.5 | 14 0.75 | 15 1 | 14 true or false: selena could choose the polynomial function f(x) = -16x² + 24x + 6 to model the data because the coefficient of determination is equal to 1. true false

Explanation:

Step1: Evaluate \( f(x) \) at \( x = 0 \)

Substitute \( x = 0 \) into \( f(x)=-16x^{2}+24x + 6 \).
\( f(0)=-16(0)^{2}+24(0)+6 = 6 \), which matches the table.

Step2: Evaluate \( f(x) \) at \( x = 0.25 \)

Substitute \( x = 0.25 \) into \( f(x) \).
\( f(0.25)=-16(0.25)^{2}+24(0.25)+6=-16(0.0625)+6 + 6=-1 + 12 = 11 \), which matches the table.

Step3: Evaluate \( f(x) \) at \( x = 0.5 \)

Substitute \( x = 0.5 \) into \( f(x) \).
\( f(0.5)=-16(0.5)^{2}+24(0.5)+6=-16(0.25)+12 + 6=-4 + 18 = 14 \), which matches the table.

Step4: Evaluate \( f(x) \) at \( x = 0.75 \)

Substitute \( x = 0.75 \) into \( f(x) \).
\( f(0.75)=-16(0.75)^{2}+24(0.75)+6=-16(0.5625)+18 + 6=-9 + 24 = 15 \), which matches the table.

Step5: Evaluate \( f(x) \) at \( x = 1 \)

Substitute \( x = 1 \) into \( f(x) \).
\( f(1)=-16(1)^{2}+24(1)+6=-16 + 24 + 6 = 14 \), which matches the table.

Since all the values from the table match the values calculated from the function \( f(x)=-16x^{2}+24x + 6 \), the data points lie perfectly on the parabola (the function). A coefficient of determination (\( R^{2} \)) of 1 means the model perfectly fits the data.

Answer:

True