Algebra Class: Equations and Functions

The equation of a line can be represented as $y - y_1 = m(x - x_1)$ where m is the slope.

The logarithm of 1 is equal to 1.

In the standard form of a quadratic equation $ax^2 + bx + c = 0$, the term c represents the horizontal intercept.

If a > 0 in a quadratic function, the graph opens downward.

The expression $X^a imes X^b = X^{a+b}$ is a property of exponents.

For an exponential function, a decay rate can be expressed as $Q = ab^t$ where b is smaller than 1.

The vertex form of a quadratic function is $a(x - h)^2 + k$ where (h,k) represents the vertex.

The formula for solving a quadratic equation includes the discriminant $b^2 - 4ac$.

Equation of a Line

One point and slope: y - y₁ = m(x - x₁), where m = slope
Two points: Slope = (y₂ - y₁)/(x₂ - x₁) , then use y - y₁ = m(x - x₁), where m = slope

Exponents

xᵃ xᵇ = x^(a+b)
xᵃ / xᵇ = x^(a-b)
(a.b)ⁿ = aⁿ . bⁿ
x⁻ⁿ = 1/xⁿ
xⁿ = 1/x⁻ⁿ
√xᵐ = x⁽ᵐ/ⁿ⁾

Solving Quadratic Equations

ax² + bx + c = 0
x = (-b ± √(b² - 4ac)) / 2a

Quadratic Expressions

Standard Form: ax² + bx + c, where c = vertical intercept
Factored Form: a(x - r)(x - s), where r and s are zeroes
Vertex Form: a(x - h)² + k, where (h, k) = vertex
If a > 0, the parabola opens up. If a < 0, the parabola opens down.

Logarithms

ln(x) = logₑ(x)
log(a.b) = log(a) + log(b)
log(a/b) = log(a) - log(b)
log(aᵗ) = t.log(a)
logₐ(a) = 1
logₐ(1) = 0
log₁₀(10ⁿ) = n
ln(eˣ) = x
logₓ(y) = z means xᶻ = y

Exponential Functions

Continuous Growth: Q = a.eᵏᵗ (where k = continuous growth rate, a = initial value)
Growth: Q = a.bᵗ (where a = initial value, b = growth factor = r + 1, r = growth rate)
Decay: Q = a.bᵗ (where a = initial value, b = remaining percentage (or fraction), b = growth factor = r + 1, r = growth rate = b - 1)

This quiz covers essential algebra concepts, including the equation of a line using points and slopes, operations with exponents, solving quadratic equations, and understanding logarithms and exponential functions. Test your knowledge and boost your skills in Algebra.