Question 189666
<h3>Vertex:</h3>


In order to find the vertex, we first need to find the x-coordinate of the vertex.



To find the x-coordinate of the vertex, use this formula: {{{x=(-b)/(2a)}}}.



{{{x=(-b)/(2a)}}} Start with the given formula.



From {{{y=4x^2-64}}}, we can see that {{{a=4}}}, {{{b=0}}}, and {{{c=-64}}}.



{{{x=(-(0))/(2(4))}}} Plug in {{{a=4}}} and {{{b=0}}}.



{{{x=(0)/(8)}}} Multiply 2 and {{{4}}} to get {{{8}}}.



{{{x=0}}} Divide.



So the x-coordinate of the vertex is {{{x=0}}}. Note: this means that the axis of symmetry is also {{{x=0}}}.



Now that we know the x-coordinate of the vertex, we can use it to find the y-coordinate of the vertex.



{{{y=4x^2-64}}} Start with the given equation.



{{{y=4(0)^2-64}}} Plug in {{{x=0}}}.



{{{y=4(0)-64}}} Square {{{0}}} to get {{{0}}}.



{{{y=0-64}}} Multiply {{{4}}} and {{{0}}} to get {{{0}}}.



{{{y=-64}}} Combine like terms.



So the y-coordinate of the vertex is {{{y=-64}}}.



So the vertex is *[Tex \LARGE \left(0,-64\right)].



<hr>

<h3>X-Intercept(s):</h3>



Remember, the x-intercepts occur when {{{y=0}}}



{{{y=4x^2-64}}} Start with the given equation.



{{{0=4x^2-64}}} Plug in {{{y=0}}}.



Notice we have a quadratic equation in the form of {{{ax^2+bx+c}}} where {{{a=4}}}, {{{b=0}}}, and {{{c=-64}}}



Let's use the quadratic formula to solve for x



{{{x = (-b +- sqrt( b^2-4ac ))/(2a)}}} Start with the quadratic formula



{{{x = (-(0) +- sqrt( (0)^2-4(4)(-64) ))/(2(4))}}} Plug in  {{{a=4}}}, {{{b=0}}}, and {{{c=-64}}}



{{{x = (0 +- sqrt( 0-4(4)(-64) ))/(2(4))}}} Square {{{0}}} to get {{{0}}}. 



{{{x = (0 +- sqrt( 0--1024 ))/(2(4))}}} Multiply {{{4(4)(-64)}}} to get {{{-1024}}}



{{{x = (0 +- sqrt( 0+1024 ))/(2(4))}}} Rewrite {{{sqrt(0--1024)}}} as {{{sqrt(0+1024)}}}



{{{x = (0 +- sqrt( 1024 ))/(2(4))}}} Add {{{0}}} to {{{1024}}} to get {{{1024}}}



{{{x = (0 +- sqrt( 1024 ))/(8)}}} Multiply {{{2}}} and {{{4}}} to get {{{8}}}. 



{{{x = (0 +- 32)/(8)}}} Take the square root of {{{1024}}} to get {{{32}}}. 



{{{x = (0 + 32)/(8)}}} or {{{x = (0 - 32)/(8)}}} Break up the expression. 



{{{x = (32)/(8)}}} or {{{x =  (-32)/(8)}}} Combine like terms. 



{{{x = 4}}} or {{{x = -4}}} Simplify. 



So the answers are {{{x = 4}}} or {{{x = -4}}} 



This means that the x-intercepts are (4,0) and (-4,0)



<hr>


<h3>Y-Intercept:</h3>




Remember, the y-intercept occurs when {{{x=0}}}



{{{y=4x^2-64}}} Start with the given equation.



{{{y=4(0)^2-64}}} Plug in {{{x=0}}}.



{{{y=4(0)-64}}} Square {{{0}}} to get {{{0}}}.



{{{y=0-64}}} Multiply {{{4}}} and {{{0}}} to get {{{0}}}.



{{{y=-64}}} Combine like terms.



So the y-intercept is (0,-64)