Problem

Find the $y$-intercept and any $x$-intercept(s) of the parabola with equation $y=-4(x-4)^{2}+16$ If the parabola doesn't have any $x$-intercepts, type DNE, meaning "does not exist." If the parabola has two $x$-intercepts, use a comma to separate them. $y$-intercept: $x$-intercept(s):

Solution

Step 1 :To find the $y$-intercept, set $x=0$ and solve for $y$ in the equation $y=-4(x-4)^{2}+16$

Step 2 :Substitute $x=0$ into the equation to get $y=-4(0-4)^{2}+16$

Step 3 :Simplify the equation to get $y=-4(-4)^{2}+16$

Step 4 :Calculate the value to get $y=-4(16)+16$

Step 5 :Simplify further to get $y=-64+16$

Step 6 :Finally, calculate the $y$-intercept to get $y=-48$

Step 7 :To find the $x$-intercept(s), set $y=0$ and solve for $x$ in the equation $y=-4(x-4)^{2}+16$

Step 8 :Substitute $y=0$ into the equation to get $0=-4(x-4)^{2}+16$

Step 9 :Rearrange the equation to get $4(x-4)^{2}=16$

Step 10 :Divide both sides by 4 to get $(x-4)^{2}=4$

Step 11 :Take the square root of both sides to get $x-4=\pm 2$

Step 12 :Solve for $x$ to get the $x$-intercept(s) as $x=4\pm 2$

Step 13 :Calculate the values to get the $x$-intercept(s) as $x=2$ and $x=6$

Step 14 :The final answer for the $y$-intercept is \(\boxed{-48}\)

Step 15 :The final answer for the $x$-intercept(s) is \(\boxed{2, 6}\)

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Source: https://solvelyapp.com/problems/ksR79wtPQE/

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