Precalculus Midterm Review: Equations, Functions, Quadratics, and Systems

Equations and Inequalities

Absolute Value Equations

Absolute value equations require splitting into two cases to solve for the variable.

• Definition: The absolute value of a number is its distance from zero on the number line, always non-negative.

• Solving |ax + b| = c: Set up two equations: ax + b = c and ax + b = -c, then solve each for x.

• Example: Solve $|4x + 12| = 20$.

  • Case 1: $4x + 12 = 20 \implies 4x = 8 \implies x = 2$

  • Case 2: $4x + 12 = -20 \implies 4x = -32 \implies x = -8$

  • Solution: $x = 2$ or $x = -8$

Completing the Square

This method rewrites a quadratic equation in the form of a perfect square trinomial to solve for x.

• Process: For $x^2 + bx = c$, add $\left(\frac{b}{2}\right)^2$ to both sides to complete the square.

• Example: Solve $x^2 + 10x = 11$ by completing the square.

  • Add $\left(\frac{10}{2}\right)^2 = 25$ to both sides: $x^2 + 10x + 25 = 11 + 25$

  • So, $(x + 5)^2 = 36$

  • Take square roots: $x + 5 = \pm6$

  • Solutions: $x = 1$ or $x = -11$

Quadratic Inequalities

Quadratic inequalities are solved by finding critical values and testing intervals.

• Steps:

  1. Set the quadratic equal to zero to find critical values.

  2. Test intervals between and beyond these values to determine where the inequality holds.

• Example: Solve $x^2 - 2x \geq 15$.

  • Rewrite: $x^2 - 2x - 15 \geq 0$

  • Factor: $(x - 5)(x + 3) \geq 0$

  • Critical values: $x = 5$, $x = -3$

  • Test intervals: $(-\infty, -3] \cup [5, \infty)$

  • Solution: $x \leq -3$ or $x \geq 5$

Complex Numbers

Complex numbers are numbers in the form $a + bi$, where $i = \sqrt{-1}$.

• Definition: $i$ is the imaginary unit, satisfying $i^2 = -1$.

• Example: Solve $3x^2 + 75 = 0$.

  • $3x^2 = -75 \implies x^2 = -25$

  • $x = \pm\sqrt{-25} = \pm5i$

  • Solution: $x = 5i$ or $x = -5i$

Functions and Their Graphs

Domain and Range

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

• Domain: Exclude values that make denominators zero or radicands of even roots negative.

• Example: Find the domain of $f(x) = \frac{1}{\sqrt{2x + 10}}$.

  • Set $2x + 10 > 0 \implies x > -5$

  • Domain: $(-5, \infty)$

Analyzing Graphs: Increasing/Decreasing Intervals

Determine where a function is increasing or decreasing by observing the graph from left to right.

• Increasing: The function rises as x increases.

• Decreasing: The function falls as x increases.

• Intervals are written using x-values.

Distance Formula

The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by:

• $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

• Example: Find the distance between (1, 5) and (-3, 2):

  • $d = \sqrt{(-3 - 1)^2 + (2 - 5)^2} = \sqrt{(-4)^2 + (-3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5$

Transformations of Functions

Transformations shift, stretch, compress, or reflect the graph of a parent function.

• General form: $g(x) = a(x - h)^2 + k$

• h: Horizontal shift (opposite sign)

• k: Vertical shift

• a: Vertical stretch/compression; if negative, reflects across the x-axis

• Example: For $h(x) = -2(x + 3)^2 - 5$:

  • Horizontal shift: Left 3 units

  • Vertical shift: Down 5 units

  • Vertical stretch by 2 and reflection across x-axis

  • Domain: $(-\infty, \infty)$

  • Range: $(-\infty, -5]$

Modeling Functions

Expressing one quantity as a function of another using known formulas.

• Example: A string of length $20x$ forms a circle. Express the area as a function of $x$.

  • Circumference: $C = 2\pi r = 20x \implies r = \frac{10x}{\pi}$

  • Area: $A = \pi r^2 = \pi \left(\frac{10x}{\pi}\right)^2 = \frac{100x^2}{\pi}$

  • Function: $A(x) = \frac{100x^2}{\pi}$

Quadratic Functions

Key Features of Parabolas

Quadratic functions have the form $f(x) = ax^2 + bx + c$ and their graphs are parabolas.

• Vertex: The turning point of the parabola.

  • x-coordinate: $x = \frac{-b}{2a}$

  • y-coordinate: Substitute x into $f(x)$

• Axis of Symmetry: The vertical line $x = h$ (where h is the x-coordinate of the vertex).

• Intercepts:

  • y-intercept: Set $x = 0$

  • x-intercepts: Set $f(x) = 0$ and solve

• Opens Up or Down: If $a > 0$, opens up; if $a < 0$, opens down.

• Example: For $f(x) = x^2 - 6x + 5$:

  • $a = 1 > 0$; opens up

  • Vertex: $x = \frac{-(-6)}{2 \cdot 1} = 3$; $y = (3)^2 - 6 \cdot 3 + 5 = 9 - 18 + 5 = -4$

  • Vertex: (3, -4)

  • Axis of symmetry: $x = 3$

  • y-intercept: $f(0) = 0^2 - 6 \cdot 0 + 5 = 5$

  • x-intercepts: $x^2 - 6x + 5 = 0 \implies (x - 5)(x - 1) = 0 \implies x = 5, 1$

Systems of Equations

Solving Systems: Substitution and Elimination

Systems of equations can be solved using substitution or elimination methods.

• Substitution Method: Solve one equation for a variable, substitute into the other.

  • Example: $2x + y = 10$, $x - y = 2$

    • From second equation: $x = y + 2$

    • Substitute into first: $2(y + 2) + y = 10 \implies 2y + 4 + y = 10 \implies 3y = 6 \implies y = 2$

    • Then $x = 2 + 2 = 4$

    • Solution: $(4, 2)$

• Elimination Method: Add or subtract equations to eliminate a variable.

  • Example: $4x - 2y = 8$, $8x - 4y = 10$

    • Multiply first equation by 2: $8x - 4y = 16$

    • Compare with second: $8x - 4y = 10$

    • Contradiction: $16 \neq 10$

    • Conclusion: System is inconsistent (no solution)

Classification of Systems

Type	Number of Solutions	Description
Consistent & Independent	One	Lines intersect at one point
Consistent & Dependent	Infinitely many	Lines are coincident (same line)
Inconsistent	None	Lines are parallel (no intersection)

Additional info: The above notes cover core precalculus topics relevant for a midterm, including equations, inequalities, functions, quadratics, and systems of equations. Practice problems are integrated as examples for each topic.