# Finding the equation of a parabola with 1 point

1. Jul 1, 2015

### Jaco Viljoen

1. The problem statement, all variables and given/known data

The sketch on the previous page shows the graph of a function f, which is a parabola with vertex R, and the graph of a function g, which is a straight line defined by g(x)=-(1/2)x. The graphs of f and g intersect at P and O(the origin). The function f is defined by:
f(x)=ax2+bx+c
Df=ℝ, Rf=[-2,∞) and b/(2a)=2.

a)Find the coordinates of R.

2. Relevant equations
g(x)=-(1/2)x
f
(x)=ax2+bx+c
Df=ℝ, Rf=[-2,∞) and b/(2a)=2.

3. The attempt at a solution
if b/(2a)=2 then my x coordinate of the vertex = 2 correct? but on the graph it looks more like -2
Is this due to the missing (-)b

I am stumped on this question, I am sure It is quite obvious but I would appreciate a point in the right direction.
Thank you,
Jaco

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• ###### Parabola.pdf
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Last edited: Jul 1, 2015
2. Jul 1, 2015

### RUber

If b/(2a) = 2, then b = 4a, so you can rewrite f(x) = ax^2 + 4ax + c. This has a minimum where 2a x +4a = 0, or where x=-2. So your original insight was right, but you might have tried to jump a logical step and missed where the negative belongs.

How can you tell what the y coordinate might be?

3. Jul 1, 2015

### RUber

Hint: f(0) = 0.

4. Jul 1, 2015

### EM_Guy

$f(x) = ax^2 + bx + c = a(x-h)^2 + k$

Given: $\frac{b}{2a} = 2$

The range of the function is from -2 to $\infty$. What does this tell you about the y-coordinate of R?

Not only should you be able to find the coordinates of R. You should be able to find the equation for the parabola (and thus find any point on the parabola).

5. Jul 3, 2015

### ehild

That is the picture:

6. Jul 6, 2015

### Jaco Viljoen

Rf={-2,∞)
so y=-2
so my vertex is (-2,-2) (Coordinates of R)
f(x)=1/2(x+2)^2-2

Last edited: Jul 6, 2015
7. Jul 6, 2015

### SammyS

Staff Emeritus
Yes, that's correct

8. Jul 6, 2015

### Jaco Viljoen

Find:
a, b and c
y=1/2(x+2)^2-2
y=1/2(x+2)(x+2)-2
y=1/2(x^2+2x+2x+4)-2
y=1/2x^2+2x+0
so a=1/2, b=2 and c=0

9. Jul 6, 2015

### SammyS

Staff Emeritus
Also is correct.

10. Jul 6, 2015

### Jaco Viljoen

Thank you Sammy,

11. Jul 6, 2015

### Jaco Viljoen

Find the equation of a line that is parallel to the straight line and passes through R.

y=-1/2x+0 point R(-2,-2)
y=mx+b
-2=-1/2x+b
-2=-1/2(-2)+b
-2=1+b
b=-3

y=-1/2x-3

12. Jul 6, 2015

### Jaco Viljoen

Find the coordinates of P
(1/2)x^2+2x+0=-(1/2)x+0
(1/2)x^2+2x+(1/2)x=0
(1/2)x^2+2&(1/2)x=0
(x^2)/2+(5x)/2=0
x^2+5x=0
x(x+5)=0

x=0 and x=-5

so (-5,?)

y=-1/2(x)
y=-1/2(-5)
y=2.5

so P (-5,2.5)

Last edited: Jul 6, 2015
13. Jul 6, 2015

### Staff: Mentor

Instead of dragging those 1/2 factors along, just multiply both sides of the equation by 2, which results in
$x^2 + 4x + x = 0$, or $x^2 + 5x = 0$.

14. Jul 6, 2015

### Jaco Viljoen

Thank you Mark,

15. Jul 6, 2015

### Jaco Viljoen

Calculate the distance between P and Q
P(-5,2.5) Q(?,0)
Find Q

x intercepts:
y=(1/2)x^2+2x+0
((1/2)x+2)(x+0)
(1/2)x=-2 and x=0
x=-4 and x=0

Q(-4,0)

d=√{(x2-x1)2+(y2-y1)2}
d=√{(-4-(-5))2+(0-2.5)2}
d=√{(-4+5))2+(0-2.5)2}
d=√{12+(-2.5)2}
d=√{1+6.25}
d=√{7.25}

The distance between P and Q is √{7.25}

Last edited: Jul 6, 2015
16. Jul 6, 2015

### Jaco Viljoen

I do not understand the following question, some help please.

Calculate the maximum vertical distance between corresponding points on the graphs of f and g on the interval[xP,0], where xP denotes the x-coordinate of P.

17. Jul 6, 2015

### Staff: Mentor

The vertical distance between the line and the parabola is $h = -\frac x 2 - \frac{x^2}{2} - 2x$. This distance function turns out also to be a quadratic, so you can find its maximum value by techniques you already know.

18. Jul 9, 2015

### Tony Mondi

hello Jaco and Sammy i have a question i can see on the equation you put 1/2 (x+2)^2-2 i am tryna find out where the 1/2 came from looking from the g=-1/2x does it mean when its parallel the -1/2 becomes positive 1/2

19. Jul 9, 2015

### Jaco Viljoen

Hi Toni,
ƒ(g) is a line,
f(x)=1/2(x+2)^2-2 is the parabola.

so no.

I used:
ƒ(x)=a(x-h)^2+k to find a, 1/2(x+2)^2-2

20. Jul 9, 2015

### Tony Mondi

thanks i got it