Chapter 9.7, Problem 30HP

To determine

To explain that graph of linear equation intersect that of the quadratic equation at vertex.

Answer to Problem 30HP

Yes, S is correct

Explanation of Solution

Given information:

S has linear equation $\mathrm{y}=\mathrm{c}$

Calculation:

Let quadratic equation

  $\mathrm{a}{\mathrm{x}}^2+\mathrm{b}\mathrm{x}+\mathrm{c}=0$

Linear equation $\mathrm{y}=\mathrm{c}$

c is constant and it is the y intercept

Then line will be parallel

Quadratic equation

  $\mathrm{a}{\mathrm{x}}^2+\mathrm{b}\mathrm{x}+\mathrm{c}=0$

a, b and c are constants

The point, where the parabola changes direction, is called the "vertex".

If the quadratic is written $\mathrm{y}={(\mathrm{x}-\mathrm{h})}^2+\mathrm{k}$ ,

Then the vertex is the point $(\mathrm{h},\mathrm{k})$ .

This makes sense, if think about it.The squared part is always positive (for a right-side-up parabola), unless it's zero. So it will always have that fixed value k , and then it will always be adding something to it to make y bigger, unless of course the squared part is zero.

So the smallest y can possibly be is $\mathrm{y}=\mathrm{k}$ , and this smallest value will happen when the squared part, $\mathrm{x}-\mathrm{h}$ , equals zero. And the squared part is zero when $\mathrm{x}-\mathrm{h}=0$ , or when $\mathrm{x}=\mathrm{h}$ .

The same reasoning works, with k being the largest value and the squared part always subtracting from it, for upside-down parabolas.

(Note: The $"\mathrm{a}"$ in the vertex form $"\mathrm{y}={(\mathrm{x}-\mathrm{h})}^2+\mathrm{k}"$ of the quadratic is the same as the $"\mathrm{a}"$ in the common form of the quadratic equation, $"\mathrm{y}=\mathrm{a}{\mathrm{x}}^2+\mathrm{b}\mathrm{x}+\mathrm{c}"$ )

For a given quadratic $\mathrm{y}=\mathrm{a}{\mathrm{x}}^2+\mathrm{b}\mathrm{x}+\mathrm{c}$ , the vertex $(\mathrm{h},\mathrm{k})$ is found by computing $\mathrm{h}=\frac{-\mathrm{b}}{2\mathrm{a}}$ , and then evaluating y at h to find k . If it have already learned the Quadratic Formula, it may find it easy to memorize the formula for k , since it is related to both the formula for h and the discriminant in the Quadratic Formula: $\mathrm{k}=\frac{4\mathrm{a}\mathrm{c}-{\mathrm{b}}^2}{4\mathrm{a}}$

Yes, S is correct