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154 coordinate, Cartesian plane, origin, $x$-axis, $y$-axis, quadrants, rectangular (Cartesian) coordinates, $x$-coordinate, abscissa, $y$-coordinate, ordinate, ordered pair

A coordinate plane, or Cartesian plane, has horizontal and vertical real number lines that intersect at the zero points of the two lines. The intersection is known as the origin.

The $x$-axis is the horizontal number line that has a positive direction to the right.

The $y$-axis is the vertical number line that has a positive direction up.

154 one-to-one correspondence between ordered pairs and points in the plane For each point in the coordinate plane, there corresponds a unique ordered pair of real numbers, and for each ordered pair of real numbers there is a unique point in the plane. 155 line segment The portion of a line between, and joining, the points $P(x_1,y_1)$ and $Q(x_2,y_2)$ in the plane. This segment is indicated by $\overline{PQ}$. 218 equivalent statements for variation 155 length of a line segment First construct a right triangle from $\overline{PQ}.$ The length of the horizontal leg of the triangle is $|x_2-x_1|$ and the length of the vertical leg is $|y_2-y_1|.$ Use the Pythagorean theorem \[ \left(\overline{PQ}\right)^2=|x_2-x_1|^2+|y_2-y_1|^2 \] to find the length. 155 distance formula \[ PQ=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2} \] 156 midpoint formula \[ \left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right) \] 157 analytic geometry Refers to using geometric figures on the coordinate axes and solving the related problems by using algebra. 161 independent variable, dependent variable, graph of an equation
  • Independent variable \[ x=-2,-1,0,1,2 \]
  • Dependent varible \[ y=2x+1 \] where $y=-3,-1,1,3,5$
  • Graph of an equation is the set of all ordered pairs $(x,y)$ that satisfy the equation.
162 point-plotting method
  1. Set up a table of values and find a few ordered pairs that satisfy the equations.
  2. Plot and label the corresponding points in the coordinate plane.
  3. Look for a pattern, and connect the plotted points to form a smooth curve.
164 symmetric with respect to the x-axis If replacing $y$ with $-y$ yields an equivalent equation. 164 symmetric with respect to the y-axis If replacing $x$ with $-x$ yields an equivalent equation. 164 symmetric with respect to the origin If replacing $x$ and $y$ with $-x$ and $-y,$ respectively, yields an equivalent equation. 167 equation of a circle in standard form \[ (x-h)^2+(y-k)^2=r^2 \] 168 equation of a circle in general form \[ x^2+y^2+Dx+Ey+F=0 \] 172 function, domain, range, output, input A function from a set $X$ to a set $Y$ is a rule of correspondance that assigns to each element $x$ in $X$ exactly one element $y$ in $Y.$ The domain of the function is the set of all numbers we choose from, and the range is the set of all number we obtain. In the equation $y=x^2,$ we can assume that the $x$-values are inputs and the $y$-values are outputs. 172 vertical line test Draw a vertical line through the graph of an equation to see if it is a function. 174 functional notation If $f$ is a function and $x$ is an input for the function, then the functional notation $f(x),$ read "$f$ of $x$", denotes the corresponding output of the function. 175 piece-wise defined function A function that can be defined by a multipart rule. \[ h(x)= \left\{ \begin{array}{ll} 1-x^2 &\text{ if } x\lt1\\ x-1 &\text{ if } 1\leq x\leq3\\ 4 &\text{ if } x\gt3.\\ \end{array} \right. \] 176 finding the domain of a function
  1. Division by zero and
  2. Even roots of negative numbers
177 graph of a function $f$ is the set of all points $(x,y)$ in the coordinate plane such that $x$ is in the domain of $f$ and $y=f(x).$ 178 tests for even and odd functions
  1. A function $f$ is even if $f(-x)=f(x)$ for every $x$ in the domain of $f.$
  2. A function $f$ is odd if $f(-x)$ for every $x$ in the domain of $f.$
179 zeros of a function The values of $x$ for which $f(x)=0.$ The real zeros of a function $f$ are the $x$-intercepts of the graph of $f.$ 184 constant function
  • \[ f(x)=k \]
  • Domain \[ (-\infty,\infty) \]
  • Range \[ \{k\} \]
184 identity function
  • \[ f(x)=x \]
  • Domain \[ (-\infty,\infty) \]
  • Range \[ (-\infty,\infty) \]
184 absolute value function
  • \[ f(x)=|x| \]
  • Domain \[ (-\infty,\infty) \]
  • Range \[ [0,\infty) \]
184 square function
  • $f(x)=x^2$
  • Domain \[ (-\infty,\infty) \]
  • Range \[ [0,\infty) \]
184 cube function
  • $f(x)=x^3$
  • Domain \[ (-\infty,\infty) \]
  • Range \[ (-\infty,\infty) \]
184 reciprocal function
  • \[ f(x)=\frac1x \]
  • Domain \[ (-\infty,0)\cup(0,\infty) \]
  • Range \[ (-\infty,0)\cup(0,\infty) \]
184 square root function
  • \[ f(x)=\sqrt{x} \]
  • Domain \[ [0,\infty) \]
  • Range \[ [0,\infty) \]
184 cube root function
  • \[ f(x)=\sqrt[3]{x} \]
  • Domain \[ (-\infty,\infty) \]
  • Range \[ (-\infty,\infty) \]
185 vertical shift $F(x)=f(x)+c$ where the graph of $f$ is shifted vertically upward $|c|$ units if $c\gt0$ or shifted downward $|c|$ units if $c\lt0.$ 185 x-intercepts, y-intercepts of a function The $x$-intercept of the graph of a function $f$ are the real zeros of the function. The $y$-intercept of the graph of a function $f$ may be found by evaluating $f(0).$ 187 horizontal shift rule \[ F(x)=f(x+c) \] where the graph of $f$ shifted horizontally to the left $|c|$ units if $c\gt0$ or shifted horizontally to the right $|c|$ units if $c\lt0.$ 189 $x$-axis reflection rule If $f$ is a function, then the graph of the function $F$ defined by \[ F(x)=-f(x) \] is the same as the graph of $ $ reflected about the $x$-axis. 189 $y$-axis reflection rule If $f$ is a function, then the graph of the function $F$ defined by \[ F(x)=f(-x) \] is the same as the graph of $f$ reflected about the $y$-axis. 191 vertical stretch and shrink rule If $f$ is a function and $c$ is a real number with $c\gt1,$ then the graph of the function $F$ defined by \[ F(x)=cf(x) \] is similar to the graph of $f$ stretched vertically by a factor of $c,$ and the graph of the funtion $G$ defined by \[ G(x)=\frac{1}{c}f(x) \] is similar to the graph of $f$ shrunk vertically by a factor of $c.$ 192 increasing function If for all $a$ and $b$ in the domain of $f,$ \[ f(a)\lt f(b)\text{ whenever } a\lt b. \] 192 decreasing function If for all $a$ and $b$ in the domain of $f,$ \[ f(a)\gt f(b)\text{ whenever } a\lt b. \] 192 constant function If for all $a$ and $b$ in the domain of $f,$ \[ f(a)=f(b) \] 197 arithmetic of functions Uses two real numbers to form a third number by the operations of addition, subtraction, multiplication, or division. 197 function sum The sum of $f$ and $g$ is the function \[ (f+g)(x)=f(x)+g(x) \] and its domain is the intersection of the domain of $f$ and the domain of $g.$ 197 function product The product of $f$ and $g$ is the function \[ (f\cdot g)(x)=f(x)\cdot g(x) \] and its domain is the intersection of the domain of $f$ and the domain of $g.$ 197 function difference The difference of $f$ and $g$ is the function \[ (f-g)(x)=f(x)-g(x) \] and its domain is the intersection of the domain of $f$ and the domain of $g.$ 197 quotient The quotient of $f$ and $g$ is the function \[ \left(\frac{f}{g}\right)(x)=\frac{f(x)}{g(x)} \] and its domain is the intersection of the domain of $f$ and the domain of $g.$ 199 composition of functions The composition of $f$ and $g$ is the function \[ (f\circ g)(x)=f(g(x)) \] and the composition of $g$ and $f$ is the function \[ (g\circ f)(x)=g(f(x)) \] 201 decomposing functions Given function $h$ that includes the two simpler functions $f$ and $g$, we look for an inner function of $g$ and an outer function of $f.$ \[ h(x)=f(g(x))=(f\circ g)(x) \] 206 inverse functions When one function "undoes" the other. For example, the cube and cube root functions. Functions $f$ and $g$ are inverses of each other if \[ f(g(x))=x \] for all $x$ in the domain of $g$ and \[ g(f(x))=x \] for all $x$ in the domain of $f.$ 205 identity function If $f$ and $g$ are inverses, then composing $f$ with $g$ in either order gives the indentity function $x$, the function that assigns each input to itself. 206 one-to-one function If, for all $a$ and $b$ in the domain of $f,$ \[ f(a)=f(b) \text{ implies } a=b \] 207 horizontal line test A function $f$ is one-to-one if no horizontal line in the coordinate plane intersects the graph of the function in more than one point. 208 finding the inverse of a function \[ f(f^{-1}(x))=f^{-1}(f(x))=x \] 210 graphs of inverse functions For every point $(a,b)$ on the graph of $g$ there corresponds a point $(b,a)$ on the graph of $g^{-1}.$ 213 applied functions Begin with an established formula and then use substitution to obtain a functional relationship between the desired variables. 216 variation Describes a functional relationship between two quantities. 216 directly proportional \[ \frac{y}{x}=k \] or \[ y=kx \] where $k$ is the constant of variation 217 inversely proportional \[ yx=k \] or \[ y=\frac{k}{x} \] where $k$ is the constant of variation 218 equivalent statements for variation
  • $y$ varies directly as the $n\text{th}$ power of $x.$ \[ y=kx^n \]
  • $y$ varies inversely as the $n\text{th}$ power of $x.$ \[ y=\frac{k}{x^n} \]
  • $y$ varies directly as $x$ and inversely as $z.$ \[ y=\frac{kx}{z} \]
  • $y$ varies jointly as $x$ and $z.$ \[ y=kxz \]