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Question: How do you change \(y = - 2{x^2} + 8x - 1\) to vertex form?...

How do you change y=−2x2+8x−1y = - 2{x^2} + 8x - 1 to vertex form?

Explanation

Solution

In the equation above, we have an equation with two variables, and we are supposed to change it to vertex form. With the help of standard vertex form for a parabola formula, we can find the vertex values, which is
y=a(x−h)2+ky = a{(x - h)^2} + k with vertex (h,k)(h,k)

Complete step-by-step solution:
Here we have an equation y=−2x2+8x−1y = - 2{x^2} + 8x - 1 and we have to convert it to vertex form which can be done with the help of the formula,
We all know that the standard vertex form for a parabola is
⇒y=a(x−h)2+k\Rightarrow y = a{(x - h)^2} + k with vertex at (h,k)(h,k)
When given a quadratic equation y=ax2+bx+cy = a{x^2} + bx + c, the xx coordinate of the vertex is, h=−b2ah = - \dfrac{b}{{2a}} and the yy coordinate of the vertex is k=a(h)2+b(h)+ck = a{(h)^2} + b(h) + c then use the form y=a(x−h)2+ky = a{(x - h)^2} + k
Explanation:
Applying the information in the answer to the given equation:
⇒h=−82(−2)\Rightarrow h = - \dfrac{8}{{2( - 2)}}
Multiplying with the bracket in denominator,
⇒h=−8−4\Rightarrow h = - \dfrac{8}{{ - 4}}
Therefore, on multiplying the negative signs and dividing the numerator with the denominator, we get,
⇒h=2\Rightarrow h = 2
Substituting all values inside the formula, we get,
⇒k=−2(2)2+8(2)−1\Rightarrow k = - 2{(2)^2} + 8(2) - 1
Substitute a=−2,h=2a = - 2,h = 2, and k=7  k = 7\;into the form:
⇒y=−2(x−2)2+7\Rightarrow y = - 2{\left( {x - 2} \right)^2} + 7

Therefore, the equation y=−2x2+8x−1y = - 2{x^2} + 8x - 1 will be converted into vertex form and be represented as y=−2(x−2)2+7y = - 2{\left( {x - 2} \right)^2} + 7

Note: The vertex form that we write is actually another form of writing out the equation of a parabola. The standard quadratic form is not really helpful while finding the vertex of a parabola. It needs to be converted to vertex form for that.
When written in "vertex form":
• (h,k)(h,k) is the vertex of the parabola, and x=hx = h is the axis of symmetry.
• The hh represents a horizontal shift (how far left, or right, the graph has shifted from x=0x = 0).
• The kk represents a vertical shift (how far up, or down, the graph has shifted from y=0y = 0).
• Notice that the hh value is subtracted in this form, and that the kk value is added.