Intermediate Algebra: Functions and Graphs

Subsection Glossary

• quadratic equation

• parabola

• extraction of roots

• compound interest

• factor

• multiplicity

• vertex

• axis of symmetry

• quadratic trinomial

• complete the square

Subsection Key Concepts

1. A quadratic equation has the standard form \(ax^2+bx=c=0\text{,}\) where \(a,~b,~\) and \(c\) are constants and \(a\) is not equal to zero.
2. The graph of a quadratic equation \(y=ax^2+bx=c\) is called a parabola. The basic parabola is the graph of \(y=x^2\text{.}\)

3. Extraction of Roots.

   To solve a quadratic equation of the form

   \begin{equation*} ax^2+c=0 \end{equation*}

   1. Isolate \(x\) on one side of the equation.
   2. Take the square root of each side.
4. Every quadratic equation has two solutions, which may be the same.
5. Solutions of a quadratic equation can be given as exact values or as decimal approximations.

6. Formulas for Volume and Surface Area.

   1. Sphere \(~~~~~~V=\dfrac{4}{3}\pi r^3~~~~~~S=4\pi r^2\)
   2. Cylinder \(~~~~~~V=\pi r^2h~~~~~~S=2\pi r^2+2 \pi rh\)
   3. Cone \(~~~~~~V=\dfrac{1}{3}\pi r^2h~~~~~~S=\pi r^2+\pi rs\)
   4. Square Pyramid \(~~~~~~V=\dfrac{1}{3}s^2h\)

7. The formula for interest compounded annually is

   \begin{equation*} A=P(1+r)^n \end{equation*}

8. Zero-Factor 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~~\text{or both} \end{equation*}

9. To Solve a Quadratic Equation by Factoring.

   1. Write the equation in standard form.
   2. Factor the left side of the equation.
   3. Apply the zero-factor principle: Set each factor equal to zero.
   4. Solve each equation. There are two solutions (which may be equal).

10. Each solution of a quadratic equation corresponds to a non-constant factor in the factored form.
11. The value of the constant \(a\) in the factored form of a quadratic equation does not affect the solutions.

12. The \(x\)-intercepts of the graph of \(y=ax^2+bx+c\) are the solutions of the equation \(ax^2+bx+c=0\text{.}\)

13. The Graph of \(y=ax^2\).

    • The parabola opens upward if \(a \gt 0\)
    • The parabola opens downward if \(a \lt 0\)
    • The magnitude of \(a\) determines how wide or narrow the parabola is.
    • The vertex, the \(x\)-intercepts, and the \(y\)-intercept all coincide at the origin.

14. The Graph of \(y=x^2+c\).

    Compared to the graph of \(y=x^2\text{,}\) the graph of \(y=x^2+c\)

    • is shifted upward by \(c\) units if \(c \gt 0\)
    • is shifted downward by \(c\) units if \(c \lt 0\)

15. For the graph of \(y=ax^2+bx+c\text{,}\) the \(x\)-coordinate of the vertex is

    \begin{equation*} x_v=\dfrac{-b}{2a} \end{equation*}

16. The square of a binomial is a quadratic trinomial,

    \begin{equation*} (x+p)^2=x^2+2px+p^2 \end{equation*}

17. To Solve a Quadratic Equation by Completing the Square.

    1. 1. Write the equation in standard form.

       2. Divide both sides of the equation by the coefficient of the quadratic term, and subtract the constant term from both sides.

    2. Complete the square on the left side:

       1. Multiply the coefficient of the first-degree term by one-half, then square the result.
       2. Add the value obtained in (a) to both sides of the equation.

    3. Write the left side of the equation as the square of a binomial. Simplify the right side.

    4. Use extraction of roots to finish the solution.