Question

How do you sketch the graph of `y=-4(x-3)^2+2` and describe the transformation?

Answer 1

See answer below.

Explanation:

If you are studying transformations and want to understand how to easily see all the effects for any function, I recommend memorising this general function. It's really worthwhile if you have exams on this stuff.

`f(x)=a(n(x-b))^k+c`

`a` is the dilation factor from the x-axis. If `a` is negative, the function is reflected in the x-axis.

`1/n` is the dilation factor from the y-axis. If `1/n` is negative, the function is reflected in the y-axis.

`b` is the horizontal translation. If `b` is positive, the function is shifted to the right, if it is negative, the function is shifted to the left.

`c` is the vertical translation. If it is positive, the function is shifted upwards, if it is negative, the function is shifted downwards.

In this case, the parent function is `y=x^2`

and it can have up to four transformations applied to it to get

`y=-4(x-3)^2+2`

Now, lets find the four parameters and describe the transformations from the parent function.

`a=-4`

The parent function is dilated by a factor of 4 from the x-axis and reflected in the x-axis.

`1/n=1`

There is no dilation effect from the y-axis.

`b=3`

The function is shifted 3 units to the right.

`c=2`

The function is shifted 2 units up.

To graph, notice that the function is already in turning point form so you can read it straight off as `(3,2)`. The parabola is upside down because `a` is negative, so now you just need to find the x and y-intercepts.

y-intercept (set `x=0`):

`y=-4(0-3)^2+2=-34`

x-intercepts (set `y=0`):

`0=-4(x-3)^2+2rArr1/2=(x-3)^2rArrx=3+-sqrt(1/2)`
graph{-4(x-3)^2+2 [-10, 10, -5, 5]}