Quadratic Functions

Question 1

What are quadratic functions?

Answer 1

eqn of form f(x) = ax^2+bx+c , where a!=0

The biggest power of x in the equation is 2.

Question 2

What are characteristic features of a quadratic functions?

Answer 2

All parabolas have an axis of symmetry and the point at which axis of symmetry intersects the parabola is called the vertex. The y-intercept of a quadratic equation is ‘c’.
Axis of symmetry will happen at x= -b/2a
Therefore coordinates of vertex is (-b/2a , f(-b/2a))

Question 3

Maximum and Minimum values of a quadratic function ?

Answer 3

The y-coordinate of the vertex of a given quadratic function is minimum or maximum value attained by the function.
1. parabola opens up and has min value, if a>0
2. parabola opens down and has maximum value, if a<0.

Question 4

What is the slope of a quadratic equation? What are the characteristics of slope?

Answer 4

Slope denotes the rate of change of y with respect to x.
Hence slope=0 will happen when the function has either maximum or minimum. (which will happen at the vertex).

Slope of a quadratic equation in std form is 2ax+b (diffentiation). we know that slope is zero at max or min points. therefore 2ax+b=0 => x = -b/2a (coordinates of the vertex) aka axis of symmetry (-b/2a, c-(b^2/4a))

Question 5

What is the vertex form of parabola?

Answer 5

y= a (x-h)^2 +k
where coordinates of vertex is (h,k)

Question 6

What is the intercept form of parabola?

Answer 6

y=a(x-p)(x-q)

Question 7

what are quadratic equations?

Answer 7

if a quadratic fn is set equal to a value, then the result is quadratic equation. If ax^2+bx+c=0 where a !=0 and a,b,c are integers, then the quadratic equation is said to be in standard form.

Question 8

What are Roots of Equations and Zeros of Functions?

Answer 8

Solutions to Quad Eqn - roots of the eqn
One method to finding roots = zeros of related quad fn. Zeros of a fn are x-intercepts of its graph and these ar ethe solutions of related equation as f(x)=0 at these points.

factorization is another method to find the solutions of the Quad Eqn

ax^2+bx+c = (ax+b) (cx+d) = acx^2 +(ad+bc)x + bd.

Question 9

What is quadratic formula?

Answer 9

we could use quadratic formula as below

(-b +/- sqrt(b^2 -4ac) )/ 2a

where b^2 -4ac is called determinant. if determinant =0 then 1 real rational root. if determinant<0 then no real root. If determinant > 0 and perfect square then there are 2 real rational roots. if determinant>0 and not perfect square then there are 2 real, irrational roots.