Study Guide: Quadratic Functions, Parabolas, and Right Triangle Trigonometry

Chapter 15: Sketching Vertical Parabolas

15.1 Review of the Algebra Related to Parabolas

Quadratic equations are closely related to the graphs of parabolas. Understanding how to solve these equations and interpret their solutions is essential for graphing and analyzing parabolas.

• Quadratic Equation: An equation of the form $ax^2 + bx + c = 0$, where $a \neq 0$.

• Factoring Quadratic Equations:

  • Without special formulas: Factor the quadratic and set each factor to zero.

  • With special formulas:

    • Difference of Squares: $a^2 - b^2 = (a + b)(a - b)$

    • Perfect Square Trinomial: $a^2 \pm 2ab + b^2 = (a \pm b)^2$

• Quadratic Formula: For $ax^2 + bx + c = 0$,

  $x = \frac{ -b \pm \sqrt{b^2 - 4ac} }{2a}$

• Discriminant: $D = b^2 - 4ac$

  • If $D > 0$ and is a perfect square: Two rational roots (factorable).

  • If $D > 0$ and not a perfect square: Two irrational roots.

  • If $D = 0$: One rational root (vertex on x-axis).

  • If $D < 0$: Two imaginary roots (no x-intercepts).

Table: The Discriminant and Solutions

Discriminant $b^2 - 4ac$	Number and Type of Solutions	Graphical Meaning
$> 0$, perfect square	Two rational solutions	Parabola crosses x-axis at two rational points
$> 0$, not perfect square	Two irrational solutions	Parabola crosses x-axis at two irrational points
$= 0$	One rational solution	Parabola touches x-axis at vertex
$< 0$	Two imaginary solutions	Parabola does not cross x-axis

15.2 Vertical Parabolas

The general equation for a vertical parabola is $y = ax^2 + bx + c$. The sign of $a$ determines the direction the parabola opens:

• If $a > 0$, the parabola opens upward.

• If $a < 0$, the parabola opens downward.

The discriminant $b^2 - 4ac$ determines the number of x-intercepts.

Key Features of Parabolas

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

• Vertex: The point $(h, k)$ where $h = -\frac{b}{2a}$ and $k = f(h)$

• Y-intercept: Set $x = 0$; $y$-intercept is $(0, c)$

• X-intercepts: Set $y = 0$ and solve for $x$

Comparison: Linear vs. Quadratic Graphs

• Linear: $y = mx + b$ (straight line, constant slope)

• Quadratic: $y = ax^2 + bx + c$ (parabola, variable slope)

Example: Sketching a Parabola

• Given $y = x^2 - 2x - 3$:

• Axis of symmetry: $x = 1$

• Vertex: $(1, -4)$

• Y-intercept: $(0, -3)$

• X-intercepts: $(-1, 0)$ and $(3, 0)$

• Additional points: $(-2, 5)$ and $(4, 5)$ (symmetrical about axis)

Steps for Sketching a Parabola

1. Find the axis of symmetry: $x = -\frac{b}{2a}$

2. Find the vertex: Substitute $x$ into the equation to get $y$

3. Find the y-intercept: Set $x = 0$

4. Find the x-intercepts: Set $y = 0$ and solve for $x$

5. Plot additional points as needed for accuracy

Special Cases

• If $b^2 - 4ac = 0$, the vertex is the only x-intercept.

• If $b^2 - 4ac < 0$, the parabola does not cross the x-axis.

Chapter 16: Right Triangle Trigonometry

16.1 The Pythagorean Theorem

In a right triangle, the sides and angles are related by the Pythagorean Theorem:

$c^2 = a^2 + b^2$

• c: Hypotenuse (longest side, opposite the right angle)

• a, b: Legs (other two sides)

Use the theorem to find any unknown side if the other two are known.

Example:

• If $a = 5$, $b = 12$, then $c = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13$

16.2 Introduction to Trigonometric Ratios

Trigonometric ratios relate the angles of a right triangle to the lengths of its sides.

• Sine: $\sin A = \frac{\text{opposite}}{\text{hypotenuse}}$

• Cosine: $\cos A = \frac{\text{adjacent}}{\text{hypotenuse}}$

• Tangent: $\tan A = \frac{\text{opposite}}{\text{adjacent}}$

Example:

• Given triangle with sides: opposite = 30, adjacent = 16, hypotenuse = 34

• $\sin A = \frac{30}{34} = \frac{15}{17}$

• $\cos A = \frac{16}{34} = \frac{8}{17}$

• $\tan A = \frac{30}{16} = \frac{15}{8}$

16.3 "Special" Right Triangles

Some right triangles have side ratios that are always the same:

• 45°-45°-90° Triangle: Both legs are equal; hypotenuse is $x\sqrt{2}$ if each leg is $x$.

• 30°-60°-90° Triangle: Shortest leg (opposite 30°) is $x$, hypotenuse is $2x$, longer leg (opposite 60°) is $x\sqrt{3}$.

Table: Trigonometric Ratios for Special Angles

Angle	$\sin \theta$	$\cos \theta$	$\tan \theta$
30°	$\frac{1}{2}$	$\frac{\sqrt{3}}{2}$	$\frac{1}{\sqrt{3}}$ or $\frac{\sqrt{3}}{3}$
45°	$\frac{\sqrt{2}}{2}$	$\frac{\sqrt{2}}{2}$	$1$
60°	$\frac{\sqrt{3}}{2}$	$\frac{1}{2}$	$\sqrt{3}$

16.4 Trigonometric Functions Using a Calculator

Scientific calculators can be used to find approximate values for sine, cosine, and tangent. Ensure the calculator is in degree mode for angle measurements in degrees.

• To find an angle given a trigonometric value, use the inverse functions: $\sin^{-1}$, $\cos^{-1}$, $\tan^{-1}$.

• Example: If $\sin A = 0.457$, then $A = \sin^{-1}(0.457) \approx 27.2^\circ$

16.5 Solving Right Triangles

To solve a right triangle means to find all sides and angles given some initial information.

1. Use the Pythagorean Theorem if two sides are known.

2. Use trigonometric ratios to find unknown sides or angles.

3. Remember: The sum of the angles in a triangle is $180^\circ$.

Example:

• Given $c = 22$, $B = 54^\circ$

• $A = 180^\circ - (54^\circ + 90^\circ) = 36^\circ$

• $a = 22 \cos 54^\circ \approx 12.9$

• $b = 22 \sin 54^\circ \approx 17.8$

16.6 Applications of Right Triangle Trigonometry

Right triangles are used to model many real-world problems, such as heights, distances, and angles of elevation or depression.

• Angle of Elevation: Angle above horizontal from observer to object.

• Angle of Depression: Angle below horizontal from observer to object.

Example Applications:

• Kite String: If 300 ft of string makes a $38^\circ$ angle with the ground, height = $300 \sin 38^\circ \approx 184.7$ ft.

• Ladder: Ladder of length $x$ reaches 9.7 m high at $75^\circ$ angle: $x = \frac{9.7}{\sin 75^\circ} \approx 10.0$ m.

• Airplane: Horizontal distance 1,500 ft at $14^\circ$ angle: path length $x = \frac{1500}{\cos 14^\circ} \approx 1546$ ft.

• Empire State Building: $x = 940 \tan 53^\circ \approx 1247$ ft tall.

• Lighthouse: $x = \frac{50}{\tan 15^\circ} \approx 187$ m from base.

Summary Table: Steps for Solving Right Triangle Problems

Given	To Find	Method
Two sides	Third side	Pythagorean Theorem
One side, one angle (not right angle)	Other sides/angles	Trigonometric ratios
Two angles	Third angle	Sum to $180^\circ$

Additional info: This guide covers the core algebra and trigonometry concepts for sketching parabolas and solving right triangle problems, including all necessary formulas, definitions, and step-by-step procedures for exam preparation.