# Questions about y = ax²+bx+c.

#### Divina

f(x) : ax²+bx+c, a≠ 0

[have to find the a(x-h)²+k form first]

Find an expression in terms of a, b, c for: (without using graph or calculator)
(i) An equation of the axis of symmetry
(ii) The maximum or minimum value
(iii) The coordinated of the vertex
(iv) The domain and the range
(v) The y-intercept of the graph of the function
(vi) The zeroes of the function

Discuss how you can predict the number of zeroes for a given quadratic function of the form y= ax²+bx+c, a≠ 0. Support the validity of your prediction with some examples.

My attempt
(ax² + bx + b/2 - b/2) + c
(ax² + bx + b/2) + c - b/2

(v) y intercept is (0,c)

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#### pencil09

f(x) : ax²+bx+c, a≠ 0

[have to find the a(x-h)²+k form first]
find$$\displaystyle a(x-h)^2+k$$form first....

$$\displaystyle ax^2+b x+c=0$$ (divided by $$\displaystyle a$$)
$$\displaystyle x^2+\frac{b}{a} x+ \frac {c}{a}=0$$(substract by $$\displaystyle \frac {c}{a}$$)

$$\displaystyle x^2+\frac{b}{a} x =-\frac {c}{a}$$ (add by $$\displaystyle (\frac {b}{2a})^2$$)

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

$$\displaystyle (x+\frac{b}{2a})^2+(\frac {c}{a}-(\frac {b}{2a})^2)=0$$

so,
$$\displaystyle h=-\frac{b}{2a}$$

$$\displaystyle k=(\frac {c}{a}-(\frac {b}{2a})^2)$$
or $$\displaystyle k=-\frac{b^2-4ac}{4a^2}$$

-----I hope it'll help U-----

#### Divina

find$$\displaystyle a(x-h)^2+k$$form first....

$$\displaystyle ax^2+b x+c=0$$ (divided by $$\displaystyle a$$)
$$\displaystyle x^2+\frac{b}{a} x+ \frac {c}{a}=0$$(substract by $$\displaystyle \frac {c}{a}$$)

$$\displaystyle x^2+\frac{b}{a} x =-\frac {c}{a}$$ (add by $$\displaystyle (\frac {b}{2a})^2$$)

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

$$\displaystyle (x+\frac{b}{2a})^2+(\frac {c}{a}-(\frac {b}{2a})^2)=0$$

so,
$$\displaystyle h=-\frac{b}{2a}$$

$$\displaystyle k=(\frac {c}{a}-(\frac {b}{2a})^2)$$
or $$\displaystyle k=-\frac{b^2-4ac}{4a^2}$$

-----I hope it'll help U-----
Thank you very much, that helped a lot