Chapter 2.2, Problem 1E

To determine

To proof: Whether a quadratic function will have a minimum/maximum value.

Explanation of Solution

Given information: In question a quadratic function is given, and it is saying whether it will have maximum or minimum value.

Formula used: We know that any quadratic function, we can write in Vertex form $y=a{(x-h)}^2+k$ and in Intercept form also $y=a(x-p)(x-q)$ .

Proof: In vertex form of parabola equation $y=a{(x-h)}^2+k$ , If $a>0$ then parabola is upward parabola, and if $a<0$ , then it shows downward parabola. Same in Intercept form of parabola equation also $y=a(x-p)(x-q)$ , if $a>0$ then parabola is upward parabola, and if $a<0$ , then it shows downward parabola. And from basic we know that upward parabola has minimum value at its vertices, and downward parabola has maximum value at its vertices. Hence we can say that if $a>0$ then parabola has minimum value, and If $a<0$ then parabola has maximum value.