Pre-Calculus: Graphing, Symmetry, and Trigonometric Identities

What are the steps to graph a function?

1. Find the zeros by setting f(x) = 0 and solving.

2. Find the y-intercept by setting x = 0.

3. Plot these points.

4. Sketch the curve using end behavior or symmetry if applicable.

What does it mean for a function to be even? Give an example.

f(-x) = f(x), symmetric about the y-axis. Example: f(x) = x².

What does it mean for a function to be odd? Give an example.

f(-x) = -f(x), symmetric about the origin. Example: f(x) = x³.

What does it mean for a function to be neither even nor odd? Example?

Does not satisfy either property. Example: f(x) = x² + x.
One X is positive, one is negative

Steps to solve Example 1.9 (linear cost function).

1. Use slope formula m = (y₂ - y₁) / (x₂ - x₁). 2. Plug slope and one point into y - y₁ = m(x - x₁). 3. Simplify to get linear equation.

Steps to solve Example 1.10 (quadratic f(x) = 3x² - 6x + 2).

1. Factor or use quadratic formula to find zeros. 2. Find vertex using x = -b/(2a). 3. Plug x into f(x) for y. 4. Sketch parabola.

Horizontal shift rule.

f(x - c) shifts right c units if c > 0, left |c| units if c < 0.

Vertical shift rule.

f(x) + c shifts up c units if c > 0, down |c| units if c < 0.

Vertical stretch rule.

If |c| > 1, g(x) = c·f(x) stretches away from x-axis.

Vertical compression rule.

If 0 < |c| < 1, g(x) = c·f(x) compresses toward x-axis.

How do you convert radians to degrees?

Multiply radians × (180/π).

cscθ in terms of sin.

cscθ = 1 / sinθ.

secθ in terms of cos.

secθ = 1 / cosθ.

cotθ in terms of sin and cos.

cotθ = cosθ / sinθ.

Reciprocal trig identities.

tanθ = sinθ / cosθ; cotθ = cosθ / sinθ; cscθ = 1 / sinθ; secθ = 1 / cosθ.

Pythagorean identities.

sin²θ + cos²θ = 1; 1 + tan²θ = sec²θ; 1 + cot²θ = csc²θ.

Addition & subtraction identities.

sin(α ± β) = sinα cosβ ± cosα sinβ; cos(α ± β) = cosα cosβ ∓ sinα sinβ.

Double-angle identities.

sin(2θ) = 2sinθcosθ; cos(2θ) = cos²θ - sin²θ = 2cos²θ - 1 = 1 - 2sin²θ.

How do you find amplitude?

Amplitude = (max - min) ÷ 2.

How do you find period of sine/cosine?

Period = 2π / b, where b is coefficient of x.

Test: amplitude, period, zeros of f(x) = sinx.

Amplitude = 1; Period = 2π; Zeros at x = nπ (n integer).

Test: amplitude, period, zeros of f(x) = cosx.

Amplitude = 1; Period = 2π; Zeros at x = π/2 + nπ (n integer).

Steps to simplify tan(sin⁻¹x).

1. Let θ = sin⁻¹x → sinθ = x. 2. Draw right triangle with opposite = x, hypotenuse = 1. 3. Adjacent = √(1 - x²). 4. tanθ = x / √(1 - x²).

How to solve natural log equations (lnx = k).

Rewrite as exponential: lnx = k → x = e^k.