We have three forms for quadratic functions (or equations):
\(\blacktriangleright \) \(S(x) = A \, x^2 + B \, x + C\) with \(A \ne 0\) : Standard Form.
\(\blacktriangleright \) \(F(t) = A \, (t - r_1)(t - r_2)\) with \(A \ne 0\) : Factored Form.
\(\blacktriangleright \) \(V(d) = A \, (d - h)^2 + k\) with \(A \ne 0\) : Vertex Form.
Vertex form comes from completing the square. It gets its name from the graph of quadratic functions.
Graphs of Quadratic Functions
Quadratic functions all have parabolas for graphs.
![[Picture]](parabolas-85e11d13ff859f41ba8b7aa0f6642a42.svg)
![[Picture]](parabolas-07589e77c2188e87e048f846eba238c8.svg)
The extreme point on a parabola is called the vertex. It is the lowest or highest point on the parabola, depending on whether the parabola opens up or down.
This can be seen from the vertex form of the formula.
The squared term, \(A \, (x - h)^2\) has the same sign as \(A\), except when it equals \(0\). That happens at \(h\). When \(x = h\), then \(V(h) = k\), which is either the minimum or maximum value of \(V\). The vertex is the graphical representation of the the extreme value of the quadratic function and where this extrema occurs in the domain.
In addition, the intercepts represent the zeros of the quadratic function and we have seen there can be \(0\), \(1\), or \(2\) real zeros for a quadratic function. Therefore, there can be \(0\), \(1\), or \(2\) intercepts for a parabola.
![[Picture]](parabolas-62596c84026e7a4998c1a178b9492ea3.svg)
\(f\) is a quadratic function, since it matches the standard form, \(A \, x^2 + B \, x + C\).
et’s get the other two forms.
Factored form: \(f(x) = (x+5)(x-1)\).
Vertex form: \(f(x) = (x+2)^2 - 9\)
Domain
\(f\) is a quadratic function, therefore its domain is \((-\infty , \infty )\).
Zeros
From the factored form, we can see that \(-5\) and \(1\) are the zeros of \(f(x)\).
Continuity
\(f\) is a quadratic function, therefore it is continuous. Quadratic functions donot have
singularities.
End-Behavior
\(f\) is a quadratic function, with a positive leading coefficient. Therefore, its
end-behavior is unbounded positively in both directions.
Behavior
\(f\) is a quadratic function, with a positive leading coefficient. Therefore, it will decreas and then increase. It switches behavior at the domain number for the vertex, which is
\(f\) decreases on \((-\infty , -2)\) and increases on \((-2, \infty )\).
Maximum and Minimum
Since \(f\) decreases on \((-\infty , -2)\) and increases on \((-2, \infty )\), we have a global minimum at \(-2\). The minimum
value is \(f(-2) = -9\). This is also a local minimum and is the only local extrema.
\(\lim \limits _{x \to -\infty } f(x) = \infty \) tells us that there is no global maximum.
Range
\(f\) is continuous, with a global maximum and unbounded positively.
The range is \([-9, \infty )\).
A Nice Graph
\(f\) is a quadratic function and will have a parabola for a graph.
From any of the three forms, we can see that the leading coefficient is \(1\), which is positive. Thus, our parabola will open up.
From the factored form, we can see that \(-5\) and \(1\) are the zeros of \(f(x)\). These will be represented by the intercepts \((-5, 0)\) and \((1,0)\).
From the vertex form, we can see that the lowest point of the parabola will be \((-2, 9)\). Or, the function \(f\) has a global minimum value of \(-9\), which occurs at \(-2\) in the domain.
![[Picture]](parabolas-025ab3d26eef7c3fea4c46b6b076a9fd.svg)
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more examples can be found by following this link
More Examples of Quadratics