Page
Description
Equation
230 linear function If $a$ and $b$ are real numbers with $a\neq0,$ then the function $F$ defined by \[ F(x)= ax+b \] is a linear function and its gragh is a straight line. 231 slope The slope $m$ of a nonvertical line through the distinct points $P(x_1,y_1)$ and $Q(x_2,y_2),$ where $x_1\neq x_2$, is \[ m=\frac{y_2-y_1}{x_2-x_1}. \] If $x_1=x_2,$ the line is vertical and its slope is undefined. 234 slope-intercept form \[ y=mx+b \] 236 general form of a linear equation \[ Ax+By+C=0 \] 237 horizontal line \[ By+C=0 \] 237 vertical line \[ Ax+C=0 \] 238 piecewise linear function If a function is defined by different linear functions on distinct subsets of its domain. 242 point-slope form of a linear equation The equation of a line having a given slope $m$ and passing through the fixed point $P(x_1,y_1)$ and an arbitrary point $Q(x,y)$ is \[ y-y_1=m(x-x_1). \] 244 intercept form of a linear equation The equation of a line whose $x$- and $y$-intercepts are $a$ and $b$, respectively, is \[ \frac{x}{a}+\frac{y}{b}=1, \text{ where } a\neq0,b\neq0. \] 245 parallel lines Two distinct nonvertical lines with slopes $m_1$ and $m_2$ are parallel lines if and only if \[ m_1=m_2 \] 247 perpendicular lines Two nonvertical lines with slopes $m_1$ and $m_2$ are perpendicular if and only if \[ m_2 = -\frac {1}{m_1} \] 248 linear depreciation In linear depreciation, the value of an asset decreases linearly over time. 252 parabola A graph that has a parabola with vertex of \[ (h,k) \] and axis of symmetry \[ x=h. \] 252 axis of symmetry, vertex Its graph is symmetric with respect to the vertical line $x=0,$ the $y$-axis. The vertical line $x=0$ is the axis of symmetry and the point where the axis of symmetry intersects the curve is the vertex. 252 quadratic function \[ F(x)=a(x-h)^2 + k \] 253 quadratic function in standard form \[ F(x)=a(x-h)^2+k \] where $a,$ $h,$ and $k$ are real numbers with $a\neq0.$ 254 quadratic function in general form \[ F(x)=ax^2+bx+c \] where $a,$ $b,$ and $c$ are real numbers with $a\neq0.$ 255 vertex formula The graph of the quadratic function \[ F(x)=ax^2+bx+c \] is a parabola with axis of symmetry \[ x=-\frac{b}{2a} \] and vertex \[ \left( \frac{b}{2a}, F\left( -\frac{b}{2a} \right) \right) \] 257 maximum and minimum of quadratic functions If $a\lt0$, then the vertex $(h,k)$ is the lowest point on the graph, referred to as its minimum value. If $a\gt0,$ then the vertex $(h,k)$ is the highest point on the graph referred to as its maximum value. 261 conic sections When a double cone is sliced with a plane, a special family of curves is formed. The curves are the circle, ellipse, parabola, and hyperbola . 262 degenerate conic section special cases, like when the plane intersects the double cone, it is possible to obtain a single point, a line, or a pair of intersecting lines, rather than the curves mentioned earlier. An equation of the form, \[ Ax^2+Bxy+Cy^2+Dx+Ey+F=0 \] 262 general quadratic equation in two unknowns \[ Ax^2+Bxy+Cy^2+Dx+Ey+F=0 \] where $A,$ $B,$ $C,$ $D,$ $E,$ and $F$ are real numbers. 263 equations of a parabola in general form The equation \[ y=ax^2+bx+c \] may appear in the form \[ Ax^2+Dx+Ey+F=0 \] where $A,$ $D,$ $E,$ and $F$ are real numbers with $A\neq0$ and $E\neq0.$ And the equation \[ x=ay^2+by+c \] may appear in the form \[ Cy^2+Dx+Ey+F=0 \] where $C,$ $D,$ $E,$ and $F$ are real numbers with $C\neq0$ and $D\neq0.$ 265 equation of an ellipse in standard form, axes of the ellipse, major axis, minor axis, vertices, center
  • Ellipse is when the circle is flatened or stretched to form an egg-shaped curve.
  • The axes of the ellipse are the vertical line segment and horizontal line segment.
  • The longer line segment is called major axis, and the shorter one is called minor axis.
  • The vertices of the ellipse are the endpoints of the major axis. The midpoint of the major/minor axis is called the center.
The equation of an ellipse in standard form with center $(h,k)$, horizontal axis of length $2a,$ and vertical axis of length $2b$ is \[ \frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1 \] 267 equation of an ellipse in general form \[ Ax^2+Cy^2+Dx+Ey+F=0 \] 268 degenerate ellipse Is the single point on the graph of the ellipse equation where the $x^2$ and $y^2$ terms have different coefficients. 269 hyperbola, asymptotes, conjugate axis, transverse axis, center
  • The graph of a hyperbola is \[ \left( \frac{x^2}{a^2} \right) -\left( \frac{y^2}{b^2} \right)=1 \] and the asymptotes of the curve are the lines \[ y=\pm\frac{a}{b}x \]
  • The line segment that connects the vertices of the hyperbola is called the transverse axis and the other axis of the hyperbola is called the conjugate axis.
  • The midpoint of both the transverse axis and the conjugate axis is the center of the hyperbola.
270 equations of a hyperbola in standard form with center $(h,k),$ horizontal transverse axis of length $2a,$ and vertical conjugate axis of length $2b$ is \[ \frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1 \] and with center of $(h,k),$ vertical transverse axis of length $2b,$ and horizontal congjugate axis of length $2a$ is \[ \frac{(y-k)^2}{b^2}-\frac{(x-h)^2}{a^2}=1 \] 272 equations of a hyperbola in general form In vertical and horizontal axes of symmetry \[ Ax^2-Cy^2+Dx+Ey+F=0 \] and \[ Cy^2-Ax^2+Dx+Ey+F=0 \] 273 degenerate hyperbola A hyperbola is not the graph of every equation of the form. When $x^2$ and $y^2$ terms have different signs, the graph of the equation is a pair of intersecting lines which is know as degenerate hyperbola. 274 graph of a general quadratic equation in two unknowns If the graph \[ Ax^2+Cy^2+Dx+Ey+F=0 \] exists and is not degenerate, then the graph is
  1. a parabola if either $A=0$ or $C=0,$ but not both.
  2. a circle if $A=C\neq0.$
  3. an ellipse if $A\neq C$ and $AC\gt0.$
  4. a hyperbola if $AC\lt0$.
277 intersection point of two lines, substitution, addition methods.

When two lines intersect at a point $P,$ one method to find the coordinates of $P$ is substitution, solving one of the equations for one of its variables.

Another method is addition, add the left- and right-hand side of the equation in order to eliminate one of the variables, and maybe multiply by one constant so the coefficient can be negative. Sometimes you might have to multiply both equations to obtain a coefficient of a variable that can be eliminated by adding the equations.

280 intersection point of a line and a parabola
  1. the line and the parabola intersect at two points.
  2. the line and the parabola intersect at one point.
  3. the line and the parabola do not intersect.
282 intersection point of other curves It can be solved by using the substitution and addition methods. 284 approximating the intersection point of two curves Sometimes it is challenging to solve the problem using substitution. Use a calculator or computer with graphing capability to find the approximate points of intersection.