Intermediate Algebra: Functions and Graphs

Subsection Glossary

• horizontal

• vertical

• parallel

• perpendicular

• conic section

• circle

• ellipse

• parabola

• hyperbola

• central conic

• major axis

• minor axis

• vertices

• covertices

• transverse axis

• conjugate axis

• asymptote

• branch

Subsection Key Concepts

1. Horizontal and Vertical Lines.

   1. The equation of the horizontal line passing through \((0, b)\) is

      \begin{gather*} y=b \end{gather*}

   2. The equation of the vertical line passing through \((a,0)\) is

      \begin{gather*} x=a \end{gather*}

2. Slopes of Horizontal and Vertical Lines.

   The slope of a horizontal line is zero.

   The slope of a vertical line is undefined.

3. Parallel and Perpendicular Lines.

   1. Two lines are parallel if their slopes are equal, that is, if

      \begin{gather*} m_1 = m_2 \end{gather*}

      or if both lines are vertical.

   2. Two lines are perpendicular if the product of their slopes is \(-1\text{,}\) that is, if

      \begin{gather*} m_1m_2 = -1 \end{gather*}

      or if one of the lines is horizontal and one is vertical.

4. The distance between two points is the length of the segment joining them.

5. Distance Formula.

   The distance \(d\) between points \(P_1(x_1, y_1)\) and \(P_2(x_2, y_2)\) is

   \begin{gather*} d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2} \end{gather*}

6. The midpoint of a segment is the point halfway between its endpoints, so that the distance from the midpoint to either endpoint is the same.

7. Midpoint Formula.

   The midpoint of the line segment joining the points \(P_1(x_1, y_1)\) and \(P_2(x_2, y_2)\) is the point \(M(\overline{x},\overline{y})\) where

   \begin{gather*} \overline{x}=\frac{x_1+x_2}{2} \quad\text{ and } \overline{y}=\frac{y_1+y_2}{2} \end{gather*}

8. A circle is the set of all points in a plane that lie at a given distance, called the radius, from a fixed point called the center.

9. Circle.

   The equation for a circle of radius \(r\) centered at the point \((h, k)\) is

   \begin{gather*} (x-h)^2+(y-k)^2=r^2 \end{gather*}

10. Ellipse.

    The standard form for an ellipse centered at \((h, k)\) is

    \begin{gather*} \frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1 \end{gather*}

11. Hyperbola.

    The equation for a hyperbola centered at the point \((h, k)\) has one of the two standard forms:

    \begin{gather*} \frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1 \end{gather*}

    \begin{gather*} \frac{(y-k)^2}{b^2}-\frac{(x-h)^2}{a^2}=1 \end{gather*}

12. Conic Sections.

    The graph of \(Ax^2+Cy^2+Dx+Ey+F=0\) is

    1. a circle of \(A=C\text{.}\)

    2. a parabola if \(A=0\) or \(C=0\) (but not both).

    3. an ellipse if \(A\) and \(C\) have the same sign.

    4. a hyperbola if \(A\) and \(C\) have opposite signs.

13. A system of two quadratic equations may have up to four solutions.