nil101 said:
I need some help with this problem please.
Use the method of completing the square to find the axis of symmetry and the maximum or minimum point of the curve
y=2+x−x2
and sketch the curve showing it min or max, and its intercepts with the x and y axes.
This is how far i've got
\(\displaystyle {
y = - (x^2 - x - 2) \cr
= - (x^2 - x + \frac{{x^2 }}{4} - 2 - \frac{{x^2 }}{4}) \cr
= - \left[ {(x - {\textstyle{x \over 2}})^2 - \frac{{8 - x^2 }}{4}} \right] \cr}\)
I make the min or max point
(2x,48−x2)
And the intercept on the y-axis
-2
I’m not sure if the above co-ordinate is the min or the max point and I don’t think the y coordinate is right. So I’m having trouble working out the x-axis intercepts or sketching the curve.
Can you help please?
Hi!
First of all, to find the y-intercept, substitute 0 for x in the original equation:
y = -x<SUB>2</SUB> + x + 2
y = - (0)<SUB>2</SUB> + 0 + 2
Solve this for y...that will be the y-intercept.
To find the x-intercept(s), subsititute 0 for y....
0 = -x<SUB>2</SUB> + x + 2
Multiply both sides by -1:
0 = x<SUB>2</SUB> - x - 2
Factor the right side:
0 = (x - 2)(x + 1)
Set each factor equal to 0 and solve for x....this will tell you where the graph crosses the x axis.
If an equation is of the form
y = ax<SUP>2</SUP> + bx + c
then its graph is a parabola. If the coefficent of x<SUP>2</SUP> is greater than or equal to 1, the parabola opens upwards, and the lowest point (minimum value of the function) occurs at the vertex. If the coefficient of x<SUP>2</SUP> is negative, then the parabola opens downward....looks like an upside-down U....and the highest point (maximum value of the function), also occurs at the vertex.
It looks like you may have been trying to complete the square, which is a good way to put the function in vertex form,
y = a(x - h)<SUP>2</SUP> + k
Then, the vertex of the parabola is the maximum or the minimum value of the function, depending on whether the parabola opens up or down.
Your function is
y = 2 + x - x<SUP>2</SUP>
Let's rearrange this:
y = - x<SUP>2</SUP> + x + 2
Completing the square is much easier if the coefficent of the x<SUP>2</SUP> term is 1. So, let's multiply both sides of the equation by -1:
-y = x<SUP>2</SUP> - x - 2
Next, let's get the terms containing x by themselves. Add 2 to both sides of the equation:
-y + 2 = x<SUP>2</SUP> - x
Divide the coefficient of x by 2, square it, and add the result to both sides of the equation. -1 divided by 2 is -1/2, and (-1/2)<SUP>2</SUP> = 1/4. Add 1/4 to both sides of the equation:
-y + 2 + (1/4) = x<SUP>2</SUP> - x + (1/4)
-y + (9/4) = (x - 1/2)<SUP>2</SUP>
-y = (x - 1/2)<SUP>2</SUP> - (9/4)
Now, multiply both sides by -1 to get 1y, or just y, on the left side:
y = -1(x - 1/2)<SUP>2</SUP> + 9/4
This equation is now in "vertex form." The vertex of the parabola is at (1/2, 9/4).
To find the maximum value of the function, substitute 9/4 for x in the original equation.
I hope this helps you.
(edited to correct careless sign errors)