We want to write the vertex form of the parabola whose vertex is (25,90) and that passes through (50,0). To do so, let's first recall the vertex form of a quadratic function.
y= a(x- h)^2+k
In this form, ( h,k) is the vertex of the parabola. Since we are given that the vertex of our function is ( 25,90), we have that h= 25 and k=90. We can use these values to partially write our equation.
y= a(x- 25)^2+90
To find the value of a, we will use the fact that the function passes through (50,0). We can substitute 50 for x and 0 for y in the above equation and solve for a.
Now that we know that a= - 18125, we can complete the equation of the function. Note that y represents height and x represents distance.
y= -18/125(x- 25)^2+90
Next, to draw the graph of the quadratic function above, we will follow five steps.
Determine and plot the intersection with the vertical axis.
Reflect the intersection with the vertical axis across the axis of symmetry.
Draw the parabola.
Identify and Plot the Vertex
Note that the function is expressed in vertex form and its vertex is already given as ( 25,90). Therefore, we can immediately plot it 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 x= h. As we have already noticed, for our function, this is h= 25. Therefore, the axis of symmetry is the line x= 25.
Determine and Plot the y-intercept
The x-coordinate of the point where the curve intercepts the y-axis is 0. Therefore, to find the y-intercept, we will substitute x=0 into the function rule and simplify.
Reflect the y-intercept 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 y-intercept is 25 units away from the axis of symmetry. Thus, there exists another point directly across the axis of symmetry that is also 25 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 downward. Since a=- 18125, this should be expected. To draw the parabola, we will connect the points with a smooth curve.
Recall that y represents height, so it cannot be negative. Therefore, we will exclude the part of the graph the curve is below the x-axis.
