Formula for Xo and Yo for graph of quadratic function

In summary, the conversation discusses the method of reducing a quadratic equation to a perfect square and how to write it in the form ##f(x)= a(X-X_0)^{2} +Y_0##. The question arises about a mistake in the last two steps, where the factor of ##a## is not accounted for. The expert explains that the factor of ##a## should be included, resulting in ##\frac{b^2}{4a^2}## instead of ##\frac{b^2}{4a}##. The conversation ends with the questioner thanking the expert for clarifying the mistake.
  • #1
Callmelucky
144
30
Homework Statement
Shouldn't it be ##\frac{b^{2}}{4a^{2}}## after raising (b/2a) to power of 2?
Relevant Equations
f(x)= ax^2 + bx + c, f(x)=a(X-Xo)^2 + Yo
Can someone please tell me where I am wrong. I am learning how to write ##a^{2} + bx + c## in this form ##f(x)= a(X-X_0)^{2} +Y_0##.
The method used in my textbook is a reduction to the perfect square. And it goes like this:

##f(x)=ax^2+bx+c##
##=a[x^2+\frac{b}{a}x]+c##
##=a\left [ x^2+\frac{b}{a}x+\left ( \frac{b}{2a} \right )^2-\left ( \frac{b}{2a} \right )^2 \right ]+c##
##=a\left ( x+\frac{b}{2a} \right )^2 -\frac{b^2}{4a}+c##
##f(x)=a\left ( x+\frac{b}{2a} \right )^2 + \frac{4ac-b^2}{4a}##

Now my question is shouldn't in the last two steps after squaring ##\frac{b}{2a}## be ##\frac{b^2}{4a^2}## and not ##\frac{b^2}{4a}## as it is written in my textbook, I have also checked on the internet but the answer is the same, although I couldn't find why.

Thank you.
 
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  • #2
You have forgotten to account for the factor of ##a## on the far left. So, ignoring the other terms you get ##a[... -(\frac{b^2}{4a^2})]... = ... -(\frac{b^2}{4a}) ...##
 
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  • #3
DaveE said:
You have forgotten to account for the factor of ##a## on the far left. So, ignoring the other terms you get ##a[... -(\frac{b^2}{4a^2})]... = ... -(\frac{b^2}{4a}) ...##
oh, yeah. Thank you.
 
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FAQ: Formula for Xo and Yo for graph of quadratic function

What is the formula for finding the x-intercept (Xo) of a quadratic function?

The formula for finding the x-intercept (Xo) of a quadratic function is Xo = (-b ± √(b^2-4ac)) / 2a, where a, b, and c are coefficients of the quadratic equation in the form ax^2 + bx + c = 0.

What is the formula for finding the y-intercept (Yo) of a quadratic function?

The formula for finding the y-intercept (Yo) of a quadratic function is Yo = c, where c is the constant term in the quadratic equation in the form ax^2 + bx + c = 0.

How do you graph a quadratic function using the formula for Xo and Yo?

To graph a quadratic function using the formula for Xo and Yo, first plot the y-intercept (Yo) on the y-axis. Then, use the x-intercept (Xo) formula to find the x-coordinate of the vertex. Plot this point on the graph. Next, use the symmetry of the parabola to plot points on either side of the vertex. Finally, connect the points to create a smooth curve.

What is the significance of the x-intercept and y-intercept in a quadratic function?

The x-intercept and y-intercept of a quadratic function represent the points where the parabola intersects the x-axis and y-axis, respectively. The x-intercept is the solution to the quadratic equation when y=0, and the y-intercept is the constant term in the quadratic equation. These points are important in graphing and analyzing quadratic functions.

Can the formula for Xo and Yo be used for all quadratic functions?

Yes, the formula for Xo and Yo can be used for all quadratic functions in the form ax^2 + bx + c = 0. However, in some cases, the quadratic equation may not have real solutions, which means there will be no x-intercepts. In these cases, the graph will not intersect the x-axis.

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