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

Feb 5, 2018

#### 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.

$f \left(x\right) = a {\left(n \left(x - b\right)\right)}^{k} + c$

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

$\frac{1}{n}$ is the dilation factor from the y-axis. If $\frac{1}{n}$ 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

$y = - 4 {\left(x - 3\right)}^{2} + 2$

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.

$\frac{1}{n} = 1$

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 $\left(3 , 2\right)$. The parabola is upside down because $a$ is negative, so now you just need to find the x and y-intercepts.

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

$y = - 4 {\left(0 - 3\right)}^{2} + 2 = - 34$

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

$0 = - 4 {\left(x - 3\right)}^{2} + 2 \Rightarrow \frac{1}{2} = {\left(x - 3\right)}^{2} \Rightarrow x = 3 \pm \sqrt{\frac{1}{2}}$
graph{-4(x-3)^2+2 [-10, 10, -5, 5]}