MATH 101.00 - Exam 4 Review Problems - Modules 10-13

Exam 4 Review Problems: Modules 10-13

Solving Quadratic Equations

• Square Root Property

  • Solve equations of the form $x^2 = k$ by taking the square root of both sides.
  • Remember to consider both positive and negative roots.
  • Examples:
    • $x^2 + 10x + 25 = 24$ can be rewritten as $(x + 5)^2 = 24$, then $x + 5 = \pm \sqrt{24}$.
    • $(x - 7)^2 = 81$ leads to $x - 7 = \pm 9$.
    • $5x^2 - 45 = 0$ simplifies to $x^2 = 9$.
    • $(5x + 2)^2 = 3$ gives $5x + 2 = \pm \sqrt{3}$.

• Completing the Square

  • Transform a quadratic equation into the form $(x + a)^2 = b$.
  • Add $\left( \frac{b}{2} \right)^2$ to both sides of the equation to complete the square.
  • Examples:
    • $x^2 - 10x + 16 = 0$ can be completed as $(x - 5)^2 = 9$.
    • $x^2 + \frac{4}{3}x - \frac{1}{3} = 0$ becomes $\left(x + \frac{2}{3}\right)^2 = \frac{5}{9}$.
    • $2x^2 + 5x - 3 = 0$ requires dividing by 2 first: $x^2 + \frac{5}{2}x - \frac{3}{2} = 0$, then complete the square.
    • $5x^2 - 20x + 10 = 0$ similarly needs division by 5: $x^2 - 4x + 2 = 0$.

• Quadratic Formula

  • Solve equations of the form $ax^2 + bx + c = 0$ using the formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.
  • Examples:
    • $2x^2 + 4x + 1 = 0$: $a = 2, b = 4, c = 1$.
    • $6x^2 + 5x - 4 = 0$: $a = 6, b = 5, c = -4$.
    • $x^2 - 6x + 25 = 0$: $a = 1, b = -6, c = 25$.
    • $5x + x^2 = -7$ should be rewritten as $x^2 + 5x + 7 = 0$: $a = 1, b = 5, c = 7$.

Constructing Quadratic Equations

• Given a solution set $\lbrace r<em>1, r</em>2 \rbrace$, the quadratic equation can be written as $(x - r<em>1)(x - r</em>2) = 0$.
  • Examples:
    • $\lbrace 3, -1 \rbrace$: $(x - 3)(x + 1) = 0$.
    • $\left\lbrace \frac{1}{2}, 4 \right\rbrace$: $\left(x - \frac{1}{2}\right)(x - 4) = 0$.
    • $\lbrace 2i, -2i \rbrace$: $(x - 2i)(x + 2i) = 0$.
    • $\lbrace 2 + \sqrt{3}, 2 - \sqrt{3} \rbrace$: $(x - (2 + \sqrt{3}))(x - (2 - \sqrt{3})) = 0$.

Graphing Quadratic Functions

• Vertex Form: $f(x) = a(x - h)^2 + k$ where $(h, k)$ is the vertex.
• Standard Form: $f(x) = ax^2 + bx + c$.
  • Vertex can be found at $x = -\frac{b}{2a}$, and then find the corresponding $y$ value.
• Intercepts
  • $y$-intercept: Set $x = 0$.
  • $x$-intercepts: Set $f(x) = 0$ and solve for $x$.
• Axis of Symmetry: Vertical line through the vertex, $x = h$ (in vertex form).
• Examples:
  • $f(x) = 2(x - 4)^2 - 3$: Vertex is $(4, -3)$.
  • $f(x) = -(x + 3)^2 + 1$: Vertex is $(-3, 1)$.
  • $f(x) = 3(x + 1)^2 - 2$: Vertex is $(-1, -2)$.
  • $f(x) = -x^2 + 5$: Vertex is $(0, 5)$.
  • $f(x) = -x^2 - 4x - 8$: Find the vertex using $x = -\frac{b}{2a}$.
  • $f(x) = x^2 + 2x - 4$
  • $f(x) = x^2 + x - \frac{3}{4}$
  • $f(x) = -2x^2 - x + 10$

Optimization Problems

• Maximizing Product: If the sum of two numbers is constant, their product is maximized when the numbers are equal.
• Revenue Maximization: Given a revenue function $R(p)$, find the price $p$ that maximizes revenue by finding the vertex of the quadratic function.
• Area Maximization: Set up an equation for the area and use the given constraints (e.g., perimeter) to express the area in terms of one variable. Then find the maximum value.
• Examples:
  • Sum of two numbers is 24: To maximize the product, both numbers should be 12.
  • $R(p) = -2.5p^2 + 500p$: Find the vertex to maximize revenue.
  • 9 feet of wood for a frame: Maximize the area of the frame.
  • 60 meters of fencing for a garden against a house: Maximize the area of the garden.
  • $h(t) = -16t^2 + 32t$: Find the maximum height of a ball thrown upward.
  • $h(t) = -4t^2 + 48t + 3$: Find the maximum height of a projectile.

Graph Transformations

• Vertical Shift: $f(x) + c$ shifts the graph up by $c$ units.
• Horizontal Shift: $f(x - c)$ shifts the graph right by $c$ units.
• Vertical Stretch/Compression: $c \cdot f(x)$ stretches (if $c > 1$) or compresses (if $0 < c < 1$) the graph vertically.
• Reflection: $-f(x)$ reflects the graph over the x-axis.
• Examples:
  • $g(x) = -|x + 2| + 3$: Transformations of f(x) = |x||$.
  • g(x) = (x - 2)^2 + 3$: Transformations of$f(x) = x^2.
  • g(x) = 2\sqrt{x + 3} - 2$: Transformations of$f(x) = \sqrt{x}.
  • g(x) = \frac{1}{2}x^3 + 4$: Transformations of$f(x) = x^3.

Piecewise Functions

• Graph each piece of the function over its specified domain.
• Pay attention to endpoints and whether they are included (closed circle) or not (open circle).
• Examples:
  • f(x) = \begin{cases} 2 & \text{if } x \leq -2 \ 2x + 3 & \text{if } x > -2 \end{cases}
  • f(x) = \begin{cases} 2x & \text{if } x < 1 \ -3 & \text{if } x \geq 1 \end{cases}
  • f(x) = \begin{cases} x^2 & \text{if } x < 3 \ 3x + 2 & \text{if } x \geq 3 \end{cases}
  • f(x) = \begin{cases} x^2 & \text{if } x \leq -3 \ 5 & \text{if } -3 < x < 4 \ x - 1 & \text{if } x \geq 4 \end{cases}

Exponential Functions

• Functions of the form f(x) = a^x$where$a > 0$and$a \neq 1.
• Domain: All real numbers.
• Range: (0, \infty)$if$a > 0.
• Graph using data points (e.g., x = -2, -1, 0, 1, 2).
• Examples:
  • f(x) = 2^x.
  • f(x) = 3^{-x}.
  • f(x) = 4^x.
  • f(x) = \left(\frac{1}{2}\right)^x.

Compound Interest

• Compound Interest Formula: A = P\left(1 + \frac{r}{n}\right)^{nt}, where:
  • A = accumulated value
  • P = principal
  • r = interest rate
  • n = number of times compounded per year
  • t = number of years
• Continuous Compound Interest Formula: A = Pe^{rt}
• Examples:
  • Investment of $3,000 at 4.5% compounded annually for 12 years.
  • Investment of $18,000 at 2% compounded quarterly for 7 years.
  • Investment of $7,200 at 6.6% compounded continuously for 3 years.
  • Investment of $2,800 at 3.9% compounded continuously for 2 years.

Function Composition

• (f \circ g)(x) = f(g(x))
• Substitute g(x)$into$f(x).
• Examples:
  • f(x) = x^2 + 2x$and$g(x) = 2x + 1$: Find$(f \circ g)(x).
  • f(x) = \sqrt{x}$and$g(x) = x - 8$: Find$(f \circ g)(x).
  • f(x) = -x$and$g(x) = x^3 + 5x^2$: Find$(g \circ f)(x).
  • f(x) = x^2$and$g(x) = 3x^2 - 4$: Find$(g \circ f)(x).

Inverse Functions

• To verify that f(x)$and$g(x)$are inverses, show that$f(g(x)) = x$and$g(f(x)) = x.
• Examples:
  • f(x) = 2x$and$g(x) = \frac{x}{2}.
  • f(x) = (x + 4)^3$and$g(x) = \sqrt[3]{x} - 4.
  • f(x) = x^2 + 7$and$g(x) = \sqrt{x - 7}.
  • f(x) = \frac{3x + 11}{x + 5}$and$g(x) = \frac{5x - 11}{3 - x}.

Logarithmic and Exponential Equations

• Exponential Form: a = \log_b c$is equivalent to$b^a = c.
• Examples:
  • 4 = \log_x 16$: Solve for$x.
  • 2 = \log_8 x$: Solve for$x.
  • 3 = \log_5 x$: Solve for$x.
  • -2 = \log_3 x$: Solve for$x.
• Logarithmic Form: b^c = a$is equivalent to$\log_b a = c.
• Examples:
  • 3^3 = 27.
  • 2^5 = 32.
  • 10^2 = 100.
  • 4^{-3} = \frac{1}{64}.

Evaluating Logarithms

• log_b a = x$means$b^x = a.
• Examples:
  • \log_3 81.
  • \log_7 7^8.
  • \log_5 \frac{1}{\sqrt{5}}.
  • \ln(e^4).

Graphing Logarithmic Functions

• Logarithmic functions are inverses of exponential functions.
• Domain: (0, \infty)$for$f(x) = \log_b x.
• Range: All real numbers.
• Vertical asymptotes at x = 0$for$f(x) = \log_b x.
• Examples:
  • f(x) = \log_3 x.
  • f(x) = \log_4 x.
  • f(x) = \log_2 (x - 2).
  • f(x) = \log_3 (x + 3).

Distance and Midpoint Formulas

• Distance Formula: d = \sqrt{(x2 - x1)^2 + (y2 - y1)^2}.
• Midpoint Formula: M = \left(\frac{x1 + x2}{2}, \frac{y1 + y2}{2}\right).
• Examples:
  • Distance between (3, 4) and (2, 2).
  • Distance between (-2, 3) and (3, -9).
  • Distance between (-5, 0) and (-2, 2).
  • Distance between (2, -3) and (5, 1).
  • Midpoint between (-8, -9) and (0, -3).
  • Midpoint between (5, 3) and (2, 9).
  • Midpoint between (4, 8) and (0, 12).
  • Midpoint between (2, 5) and (8, 3).

Circle Equations

• Standard Form: (x - h)^2 + (y - k)^2 = r^2$, where$(h, k)$is the center and$r is the radius.
• Examples:
  • Center (2, 3) and radius 4: Write the standard form equation.
  • Given (x - 4)^2 + (y + 3)^2 = 25$$: Find the center and radius.