Question

How to plot quadratic when given the turning point and one of the x intercepts?

Can someone please explain to me how to do question 2? Maybe just pick one of them to explain,so I have a big of idea of what to do next. Thank you so much!

Answer 1

See explanation...

Explanation:

If the parabola is upright - as these examples are - then it will be laterally symmetrical about its axis, which is the vertical line through the vertex.

So for example, given (2a):

• Vertex at `(2, -6)`
• One `x` intercept at `6`

The axis will be `x=2`, so the given `x` intercept is `4` units to the right of the axis.

So the other `x` intercept will be `4` units to the left of the axis at `-2`.

graph{(y - 3/8(x-6)(x+2))((x-2)^2+(y+6)^2-0.01)((x-6)^2+y^2-0.01)((x+2)^2+y^2-0.01)(x-2) = 0 [-7.58, 12.42, -7.36, 2.64]}

`color(white)()`
General case

Given:

• Vertex at `(h, k)`
• `x` intercept at `x_1`

Then the axis will be `x = h` and the other `x` intercept:

`x_2 = h - (x_1-h) = 2h-x_1`

The equation of the parabola may be written in vertex form as:

`y = a(x-h)^2+k`

for some constant `a` (to be determined).

The given `x` intercept `(x_1, 0)` satisfies this equation, so:

`0 = a(x_1-h)^2+k`

and hence:

`a = -k/(x_1-h)^2`

Note that this is undefined in the case where `x_1 = h`. That is, when the vertex is an `x` intercept, resulting in an indeterminate value for `a` (any value would result in a parabola satisfying the conditions).

Provided `x_1 != h` we can write:

`y = -k/(x_1-h)^2(x-h)^2+k`

`color(white)()`
Application

As an example of how you might apply the results we found for the general case, let's go back to exercise 2a...

• Vertex `(h, k) = (2, -6)`

• One `x` intercept at `x_1 = 6`

The other `x` intercept is:

`x_2 = 2h-x_1 = 2(color(blue)(2))-color(blue)(6) = 4-6 = -2`

The constant for the vertex form of the equation is:

`a = -k/(x_1-h)^2 = -(color(blue)(-6))/(color(blue)(6)-color(blue)(2))^2 = 6/4^2 = 6/16 = 3/8`

The equation of the parabola can be written:

`y = a(x-h)^2+k`

That is:

`y = 3/8(x-2)^2-6`