Rewrite the equation in vertex form. name the vertex and y-intercept.
y = x^2 – 10x + 15
y = (x – 10)^2 –10
vertex: (–10, –10)
y-intercept: (0, –10)
y = (x – 5)^2 – 10
vertex: (5, –10)
y-intercept: (0, 15)
y = (x – 5)^2 + 40
vertex: (5, –10)
y-intercept: (0, 15)
y = (x – 10)^2 + 20
vertex: (–10, –10)
y-intercept: (0, –10)

1 number

if we use a graphing calculator, there is only one such number that satisfies the conclusion of the mean value theorem, that is, a number such that

where the right-hand side is the average rate of change of f on the interval [0.5,0.75], which is the slope of the secant line.

in the attached graph, the black line is the secant line which has its slope as the average rate of change of f on [0.5,0.75], while the orange line is the only tangent line to the graph of f in the interval that satisfies the mvt conclusion.