A quadratic function has the form \(f (x) = ax^2 + bx + c\text{,}\) where \(a\text{,}\) \(b\text{,}\) and \(c\) are constants and \(a\) is not equal to zero.
ZeroFactor Principle.
The product of two factors equals zero if and only if one or both of the factors equals zero. In symbols,
\begin{equation*}
ab=0 ~~\text{ if and only if }~~ a=0 ~~\text{ or }~~ b=0
\end{equation*}
The \(x\)intercepts of the graph of \(y = f (x)\) are the solutions of the equation \(f (x) = 0\text{.}\)

A quadratic equation written as \(ax^2 +bx+c=0\) is in standard form.
A quadratic equation written as \(a(x  r_1 )(x  r_2)=0\) is in factored form.
To Solve a Quadratic Equation by Factoring.
Write the equation in standard form.
Factor the left side of the equation.
Apply the zerofactor principle: Set each factor equal to zero.
Solve each equation. There are two solutions (which may be equal).
Every quadratic equation has two solutions, which may be the same.
The value of the constant \(a\) in the factored form of a quadratic equation does not affect the solutions.
Each solution of a quadratic equation corresponds to a factor in the factored form.
An equation is called quadratic in form if we can use a substitution to write it as \(au^2 + bu + c = 0\text{,}\) where \(u\) stands for an algebraic expression.
The square of the binomial is a quadratic trinomial,
\begin{equation*}
(x+p)^2 =x^2 +2px+p^2
\end{equation*}
To Solve a Quadratic Equation by Completing the Square.
Write the equation in standard form.
Divide both sides of the equation by the coefficient of the quadratic term, and subtract the constant term from both sides.
Complete the square on the left side:
Multiply the coefficient of the firstdegree term by onehalf, then square the result.
Add the value obtained in (a) to both sides of the equation.
Write the left side of the equation as the square of a binomial. Simplify the right side.
Use extraction of roots to finish the solution.
The Quadratic Formula.
The solutions of the equation \(ax^2 + bx + c = 0\text{, }~~~ a \ne 0\text{,}\) are
\begin{equation*}
\blert{x=\frac{b \pm \sqrt{b^2  4ac}}{2a}}
\end{equation*}
We have four methods for solving quadratic equations: extracting roots, factoring, completing the square, and using the quadratic formula. The first two methods are faster, but they do not work on all equations. The last two methods work on any quadratic equation.
The graph of a quadratic function \(f (x) = ax^2 + bx + c\) is called a parabola. The values of the constants \(a\text{,}\) \(b\text{,}\) and \(c\) determine the location and orientation of the parabola.

For the graph of \(y = ax^2 + bx + c\text{,}\) the \(x\)coordinate of the vertex is \(x_v = \dfrac{b}{2a}\text{.}\)
To find the \(y\)coordinate of the vertex, we substitute \(x_v\) into the formula for the parabola.
The graph of the quadratic function \(y = ax^2 + bx + c\) may have two, one, or no \(x\)intercepts, according to the number of distinct realvalued solutions of the equation \(ax^2 + bx + c = 0\text{.}\)
The Discriminant.
The discriminant of a quadratic equation is \(D = b^2  4ac\text{.}\)
If \(D \gt 0\text{,}\) there are two unequal real solutions.
If \(D = 0\text{,}\) there is one real solution of multiplicity two.
If \(D \lt 0\text{,}\) there are two complex solutions.
To Graph the Quadratic Function \(f(x) = ax^2 + bx + c\text{:}\).
Determine whether the parabola opens upward (if \(a \gt 0\)) or downward (if \(a \lt 0\)).
Locate the vertex of the parabola.
The \(x\)coordinate of the vertex is \(x_v =\dfrac{b}{2a}\text{.}\)
Find the \(y\)coordinate of the vertex by substituting \(x_v\) into the equation of the parabola.
Locate the \(x\)intercepts (if any) by setting \(y = 0\) and solving for \(x\text{.}\)
Locate the \(y\)intercept by evaluating \(y\) for \(x = 0\text{.}\)
Locate the point symmetric to the \(y\)intercept across the axis of symmetry.
Quadratic models may arise as the product of two variables.
The maximum or minimum of a quadratic function occurs at the vertex.
Vertex Form for a Quadratic Function.
A quadratic function \(y = ax^2 + bx + c\text{,}\) \(a \ne 0\text{,}\) can be written in the vertex form
\begin{equation*}
y = a(x  x_v)^2 + y_v
\end{equation*}
where the vertex of the graph is \((x_v, y_v)\text{.}\)
We can convert a quadratic equation to vertex form by completing the square.
We can graph a quadratic equation in vertex form using transformations.
A \(2\times 2\) system involving quadratic equations may have one, two, or no solutions.
We can use a graphical technique to solve quadratic inequalities.
To Solve a Quadratic Inequality Algebraically:.
Write the inequality in standard form: One side is \(0\text{,}\) and the other has the form \(ax^2 + bx + c\text{.}\)
Find the \(x\)intercepts of the graph of \(y = ax^2 + bx + c\) by setting \(y = 0\) and solving for \(x\text{.}\)
Make a rough sketch of the graph, using the sign of \(a\) to determine whether the parabola opens upward or downward.
Decide which intervals on the \(x\)axis give the correct sign for \(y\text{.}\)
We need three points to determine the equation of a parabola.
We can use the method of elimination to find the equation of a parabola through three points.
If we know the vertex of a parabola, we need only one other point to find its equation.
We can use quadratic regression to fit a parabola to a collection of data points.
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