Precalculus Study Guide: Analytic Geometry, Functions, and Polynomial Analysis

Analytic Geometry

Distance and Midpoint Formulas

Analytic geometry provides tools for calculating distances and midpoints between points in the coordinate plane.

• Distance Formula: The distance $d$ between points $P_1 = (x_1, y_1)$ and $P_2 = (x_2, y_2)$ is given by: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

• Midpoint Formula: The midpoint $M$ of the segment joining $P_1$ and $P_2$ is: $M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$

• Example: For $P_1 = (-2, 4)$ and $P_2 = (7, 0)$: Distance: $d = \sqrt{(7 - (-2))^2 + (0 - 4)^2} = \sqrt{81 + 16} = \sqrt{97}$ Midpoint: $M = \left( \frac{-2 + 7}{2}, \frac{4 + 0}{2} \right) = (2.5, 2)$

Equations of Circles

The equation of a circle describes all points at a fixed distance (radius) from a center point.

• Standard Form: $(x - h)^2 + (y - k)^2 = r^2$

• General Form: $x^2 + y^2 + Dx + Ey + F = 0$

• Example: Center $(-6, 8)$, radius $10$: Standard:$(x + 6)^2 + (y - 8)^2 = 100$General:$x^2 + y^2 + 12x - 16y + 0 = 100$ (expand and rearrange)

• Intercepts: Set $x=0$ or $y=0$ and solve for the other variable.

• Tangent Line: A tangent to the circle at a point touches the circle at exactly one point. If tangent to the y-axis, the radius equals the x-coordinate of the center.

Functions and Their Properties

Domain and Range

The domain of a function is the set of all possible input values (x-values), and the range is the set of all possible output values (y-values).

• Example: For $f(x) = \frac{4}{3x - 27}$, the domain excludes $x = 9$ (where denominator is zero): Domain: $(-\infty, 9) \cup (9, \infty)$

Function Transformations

Transformations include shifting, stretching, compressing, and reflecting graphs.

• Vertical Shift: $f(x) + k$ shifts up/down by $k$ units.

• Horizontal Shift: $f(x + h)$ shifts left/right by $h$ units.

• Reflection: $-f(x)$ reflects over the x-axis; $f(-x)$ reflects over the y-axis.

• Stretch/Compression: $af(x)$ stretches vertically by $a$; $f(bx)$ compresses horizontally by $b$.

• Example: $g(x) = \sqrt{x - 2}$ is a right shift of $\sqrt{x}$ by 2 units.

Even and Odd Functions

A function is even if $f(-x) = f(x)$ for all $x$ in the domain, and odd if $f(-x) = -f(x)$.

• Example: $g(x) = -3 - 2$ is neither even nor odd.

One-to-One Functions and Inverses

A function is one-to-one if each output is produced by exactly one input. The horizontal line test is used to determine this from a graph.

• Inverse Function: If $f$ is one-to-one, its inverse $f^{-1}$ satisfies $f(f^{-1}(x)) = x$.

• Example: For $f(x) = x^3$, $f^{-1}(x) = \sqrt[3]{x}$.

Quadratic Functions

Vertex, Axis of Symmetry, and Concavity

The graph of a quadratic function $f(x) = ax^2 + bx + c$ is a parabola.

• Vertex: $\left( -\frac{b}{2a}, f\left(-\frac{b}{2a}\right) \right)$

• Axis of Symmetry: $x = -\frac{b}{2a}$

• Concavity: If $a > 0$, parabola opens up (concave up); if $a < 0$, opens down (concave down).

• Example: $f(x) = x^2 + 4x - 12$ Vertex: $(-2, -16)$ Axis: $x = -2$ Concave up (since $a = 1 > 0$)

Intercepts

• Y-intercept: Set $x = 0$; $f(0) = c$

• X-intercepts: Solve $ax^2 + bx + c = 0$ using the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Domain, Range, and Intervals of Increase/Decrease

• Domain: $(-\infty, \infty)$ for all quadratics

• Range: $[k, \infty)$ if concave up; $(-\infty, k]$ if concave down, where $k$ is the vertex y-value

• Increasing/Decreasing: Decreasing on $(-\infty, h)$, increasing on $(h, \infty)$, where $h$ is the vertex x-value

Maximum and Minimum Values

• Maximum: If $a < 0$, vertex is maximum

• Minimum: If $a > 0$, vertex is minimum

Quadratic Inequalities

• Solve $ax^2 + bx + c < 0$ or $> 0$ by finding roots and testing intervals.

Polynomial Functions

Definition and Degree

A polynomial function is an expression of the form $f(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_0$ where $n$ is a non-negative integer.

• Degree: The highest power of $x$ with a nonzero coefficient.

• Leading Term: The term with the highest degree.

• Constant Term: The term without $x$.

Zeros and Multiplicity

• Zero: Value $x$ where $f(x) = 0$

• Multiplicity: Number of times a zero is repeated

• Graph Behavior: If multiplicity is odd, graph crosses x-axis; if even, graph touches x-axis.

End Behavior and Turning Points

• End Behavior: Determined by leading term $a_nx^n$

• Maximum Number of Turning Points: $n - 1$ for degree $n$

Polynomial Construction

• Given zeros and degree, construct polynomial by multiplying factors $(x - r)$ for each zero $r$.

• Example: Zeros $-2$ (multiplicity 1), $-3$ (multiplicity 2), degree 3: $f(x) = a(x + 2)(x + 3)^2$

Remainder and Factor Theorems

• Remainder Theorem: The remainder of $f(x)$ divided by $(x - c)$ is $f(c)$.

• Factor Theorem: $(x - c)$ is a factor of $f(x)$ if and only if $f(c) = 0$.

Rational Zeros Theorem

• Possible rational zeros are $\pm$ factors of constant term over factors of leading coefficient.

Rational Functions

Domain and Asymptotes

• Domain: All real numbers except where denominator is zero.

• Vertical Asymptotes: Values of $x$ where denominator is zero and numerator is nonzero.

• Horizontal Asymptotes: Determined by degrees of numerator and denominator:

  Degree (Numerator)	Degree (Denominator)	Horizontal Asymptote
  Less than	Greater than	$y = 0$
  Equal	Equal	$y = \frac{a_n}{b_n}$
  Greater than	Less than	None (may have oblique)

• Oblique Asymptotes: Occur when degree of numerator is exactly one more than denominator; found by polynomial division.

Exponential and Logarithmic Functions

Exponential Functions

• Form: $f(x) = a b^x$

• Domain: $(-\infty, \infty)$

• Range: $(0, \infty)$ for $a > 0$

• Horizontal Asymptote: $y = 0$

Logarithmic Functions

• Form: $f(x) = \log_b(x)$

• Domain: $(0, \infty)$

• Range: $(-\infty, \infty)$

• Vertical Asymptote: $x = 0$

• Properties:

  • $\log_b(xy) = \log_b(x) + \log_b(y)$

  • $\log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y)$

  • $\log_b(x^k) = k \log_b(x)$

Solving Exponential and Logarithmic Equations

• Isolate the exponential or logarithmic term, then use logarithms or exponentials to solve.

• Example: $\log(x) + \log(x-3) = 1$ Combine: $\log(x(x-3)) = 1$ So $x(x-3) = 10$

Applications: Exponential Decay

• Decay Model: $A(t) = A_0 e^{kt}$, where $k < 0$

• Half-life: $t_{1/2} = \frac{\ln 2}{|k|}$

• Example: For strontium-90, $A(t) = 800 e^{-0.024t}$

Summary Table: Key Formulas

Concept	Formula
Distance	$\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
Midpoint	$\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$
Circle (Standard)	$(x - h)^2 + (y - k)^2 = r^2$
Quadratic Vertex	$\left( -\frac{b}{2a}, f\left(-\frac{b}{2a}\right) \right)$
Quadratic Formula	$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
Exponential	$A(t) = A_0 e^{kt}$
Half-life	$t_{1/2} = \frac{\ln 2}{|k|}$
Logarithm Product	$\log_b(xy) = \log_b(x) + \log_b(y)$

Additional info:

• Some questions reference graphing tools and graphical analysis; students should practice sketching graphs and identifying key features such as intercepts, asymptotes, and intervals of increase/decrease.

• For inequalities, always test intervals between roots to determine where the inequality holds.

• For polynomial construction, ensure the degree matches the sum of multiplicities.