Quadratic Functions (Parabolas) Formulas and Notes

Quadratic Functions(Parabolas)

---

These are functions in the form \displaystyle y=ax^{2}+bx+c.
a cannot be zero. c is the y-intercept. A positive a value produces a graph that opens upwards. A negative a value produces a graph that opens downwards.
Turning Point: The x coordinate of the value is \displaystyle x= \frac{-b}{2a}
Example
\displaystyle f(x)=x^{2}
\displaystyle a=1, b=0, c=0
(c=0: the graph cuts the y-axis at 0)

Turning Point: x-value - \displaystyle x= \frac{-b}{2a} = \frac{-(0)}{2(1)} = 0

Turning Point: y-value - \displaystyle f(0) = (0)^{2} = 0

Turning Point: \displaystyle (0,0)

x	-2	-1	0	1	2
f(x)	4	1	0	1	4

Example
\displaystyle f(x)=2x^{2}+3x-4
\displaystyle a=2, b=3, c=-4
a is positive, the graph will open upwards

Y Intercept: (c=-4: the graph cuts the y-axis at -4)

X Intercepts: let f(x)=0 , the solve for x: \displaystyle 2x^{2}+3x-4 = 0

\displaystyle x = -2.351 , x = 0.851

Turning Point: x-value - \displaystyle x= \frac{-b}{2a} = \frac{-(3)}{2(2)} = - \frac{3}{4}

Turning Point: y-value - \displaystyle f \left( - \frac{3}{4} \right) = 2 \left( - \frac{3}{4} \right)^{2} + 3 \left( - \frac{3}{4} \right) - 4 = -5.125

Turning Point: \displaystyle (- \frac{3}{4}, -5.125)

x	-2	-1	0	1	2
f(x)	-2	-5	-4	1	10

Example
\displaystyle f(x)=-2x^{2}-3x+4
\displaystyle a=-2, b=-3, c=4
a is negative, the graph will open downwards

Y Intercept: (c=4: the graph cuts the y-axis at 4)

X Intercepts: let f(x)=0 , the solve for x: \displaystyle -2x^{2}-3x+4 = 0

\displaystyle x = -2.351 , x = 0.851

Turning Point: x-value - \displaystyle x= \frac{-b}{2a} = \frac{-(-3)}{2(-2)} = - \frac{3}{4}

Turning Point: y-value - \displaystyle f \left( - \frac{3}{4} \right) = -2 \left( - \frac{3}{4} \right)^{2} + 3 \left( - \frac{3}{4} \right) - 4 = 5.125

Turning Point: \displaystyle (- \frac{3}{4}, 5.125)

x	-2	-1	0	1	2
f(x)	-2	-5	-4	1	10