Question

The graph of $y=(2x-4)(x-4)$ is a parabola in the $xy-$plane. In which of the following equivalent equations do the $x$- and $y$-coordinates of the vertex of the parabola appear as constants or coefficients?

• A. $y=2x^2-12x+16$
• B. $y=2x(x-6)+16$
• C. $y=2{(x-3)}^2-2$
• D. $y=(x-2)(2x-8)$

Answer 1

Answer: (C)

$y=2{(x-3)}^2-2$

Choice $A$ is not the correct answer. This answer displays the location of the $y$-value of the $y$-intercept of the graph $(0,16)$ as a constant.

Choice $B$ is not the correct answer. This answer displays the location of the $y$-value of the $y$-intercept of the graph open parentheses $(0,16)$ as a constant.

Choice $C$ is correct. The equation $y=(2x-4)(x-4)$ can be written in vertex form, $y=a{(x-h)}^2+k$ to display the vertex,$(h,k)$ of the parabola. To put the equation in vertex form, first multiply:

$(2x-4)(x-4)$

$=2x^2-8x-4x+16$

$=2x^2-12x+16=2(x^2-6x+8)=0$

$\Rightarrow x^2-2\times 3x+9-9+8=0$

$\Rightarrow {(x-3)}^2-1=0$

$\Rightarrow x-3=\pm 1$

$\therefore x=3\pm 1=4,2$

$y=0$ for $x=2,4$

Choice $D$ is not the correct answer. This answer displays the location of the $x$-value of one of the $x$-intercepts of the graph $(2,0)$ as a constant.