Diagram shows the curve of a quadratic function $\displaystyle f(x) = ax^2 + bx + c$, where $\displaystyle a$, $\displaystyle b$ and $\displaystyle c$ are constant.

a)find the value of $\displaystyle a$,$\displaystyle b$ and $\displaystyle c$

b)Determine the range value of $\displaystyle x$ if $\displaystyle f(x) \geq 1$

2. Originally Posted by mastermin346
Diagram shows the curve of a quadratic function $\displaystyle f(x) = ax^2 + bx + c$, where $\displaystyle a$, $\displaystyle b$ and $\displaystyle c$ are constant.

a)find the value of $\displaystyle a$,$\displaystyle b$ and $\displaystyle c$

b)Determine the range value of $\displaystyle x$ if $\displaystyle f(x) \geq 1$
From the point (0, -2): -2 = c.

Update the model: y = ax^2 + bx - 2.

From the point (-1, -3): -3 = a - b - 2.

From the turning point (-1, -3): -1 = -b/(2a).

Solve simultaneously for a and b.

(b) cannot be done until you have correctly answered (a).

3. The point at which the curve crosses the 'y' axis is the value of c. This being -2.
by using the complete the square form, we can work out the rest of the equation.
To do this, we use the vertex, or lowest point of the graph.
y=(x+1)^2-3
this gives us x^2+2x+1-3 or x^2+2x-2
Thus. The values are ....

I cannot recall an answer to part b. But I hope this has helped

4. hi!

$\displaystyle f(x) = ax^2 + bx + c$

$\displaystyle =ax^2 + bx + \left(\frac{b}{2}\right)^2 - \left(\frac{b^2}{4}\right) +c$

$\displaystyle =a\left(x + \frac{b}{2}\right)^2 - \frac{b^2}{4} + c$

$\displaystyle =a\left(x + \frac{b}{2}\right)^2 + \left(-\frac{b^2}{4} + c \right)$

$\displaystyle x + \frac{b}{2} = 0$

$\displaystyle -1 + \frac{b}{2} = 0$

$\displaystyle b= 2$

minimum value = $\displaystyle -3$

$\displaystyle -\frac{b^2}{4} + c = -3$

$\displaystyle c=-2$

the value of $\displaystyle a$ is

$\displaystyle f(x) = ax^2 + bx + c$

at point $\displaystyle (-1,-3)$

$\displaystyle -3 = a + 2(-1) - 2$

$\displaystyle -3 = a - 4$

$\displaystyle a = 1$

is correct sir?

5. Seems correct to me.

You can always double check by putting in x=-1 and seeing if the y co-ordinate is correct.

7. $\displaystyle x^2 + 2x - 2 \geq 1$

$\displaystyle x^2 + 2x - 3 \geq 0$

Can you carry on?

8. so,the range is $\displaystyle f(x) \geq 1$ and $\displaystyle f(x) \leq -3$?

9. Yep.