Note that the function is expressed in vertex form.
h(t)=5(t-2.5)^2 ⇔ h(t)=5(t-2.5)^2+0
Let's compare the general formula for the vertex form with our function.
Formula:& h(t)= a(t- h )^2 +k
Function:& h(t)= 5(t- 2.5)^2+
We can see that a= 5, h= 2.5, and k= . Recall that the vertex of a quadratic function written in vertex form is the point ( h,k). Therefore, the vertex of the given equation is ( 2.5, ). Let's plot the vertex on a coordinate plane.
Draw the Axis of Symmetry
The axis of symmetry of a quadratic function written in vertex form is the vertical line with equation t= h. As we have already noticed, for our function, this is h= 2.5. Therefore, the axis of symmetry is the line t= 2.5.
Determine and Plot the y-intercept
Recall that the first coordinate of the point where the parabola intercepts the vertical axis is 0. Therefore, to find the its second coordinate, we will substitute t=0 in the function.
The intersection of the parabola and the vertical axis occurs at the point (0,31.25).
Reflect the Intersection With the Vertical Axis Across the Axis of Symmetry
The axis of symmetry divides the parabola into two mirror images. Therefore, points on one side of the parabola can be reflected across the axis of symmetry. Notice that the intersection with the vertical axis is 2.5 units away from the axis of symmetry. Thus, there exists another point directly across the axis of symmetry that is also 2.5 units away, in the opposite direction.
Draw the parabola
Now, with three points plotted, the general shape of the parabola can be seen. It appears that the parabola faces upward. Since a=5 in the given function rule, this should be expected. To draw the parabola, we will connect the points with a smooth curve. Note that negative values for t do not make sense. We will draw the curve for t≥ 0.
b Given that r(t)= 2h(t), and 2 is greater than 1, we can tell that the graph of r(t) is a vertical stretch by a factor of 2 of the graph of h(t). To draw its graph, we will multiply the second coordinate of the points on h(t) by 2.
Now, we can remove the unnecessary parts and have the graph of r(t).
c Recall that t is the number of seconds after beginning the dive. Therefore, to determine which bird starts its dive from a greater height, we will compare their height at t=0. This means that we should compare the intersection with the vertical axis of both graphs.
As we can see above, the intersection with the vertical axis of the graph of r(t) is greater than the intersection with the vertical axis of the graph of h(t). Therefore, the bird whose dive is represented by r(t) starts its dive from a greater height.
