Standard form of quadratic function is f(x)=
.
a, b, c are real numbers where
.
Vertex of such a quadratic functions is (p, q) = (
))
Direction of opening depends on
again. If a>0 then direction of opening is upwards and if a<0 then direction of opening is downwards.
Example, we have quadratic function f(x)=
a = 1, b = 2 and c = 1
Therefore, vertex = (p ,q) = (-1, 0)
Because, a>0 we will have direction of opening of graph upwards.
Once we have value of p and q, we can easily find maximum or minimum values, axis of symmetry, domain, range etc. Refer to article Quadratic Functions in Vertex Form.
y-intercept: Putting x=0 in
, we get
Therefore, function
intersects at (0,c) at y-axis.
Example we have
y-intercept = c = 1. It means that graph of the function will intersect at (0,1) at y-axis.
Quadratic function can have exactly 1 y-intercept.
x-intercept: To find x-intercept of the function
, we need to put y=0, we get
, it forms a quadratic equation. Therefore, to find x-intercepts we need to solve this quadratic equation.
We can use different strategies like factorization of quadratic equations, completing square of quadratic equations and quadratic formula to solve quadratic equations. We will use quadratic formula here to solve one example.
According to quadratic formula we have
Example, we have quadratic function
and we want to find its x-intercepts. Putting y=0, we get
a = 1, b =4 and c = 3 in this case.
According to quadratic formula, we have
,
Therefore, we have 2 x-intercepts (-1,0) and (-3,0) for quadratic function
.
Quadratic function can have 0, 1 or 2 x-intercepts.
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