SAT Math — Algebra & Functions

Discriminant of a quadratic equation

$b^2 - 4ac$

What does the discriminant tell us?

The discriminant indicates the nature of the roots: - Positive: two real and distinct roots - Zero: one real repeated root - Negative: two complex roots

Factoring a quadratic expression

Expressing it as a product of two binomials, e.g., $ax^2 + bx + c = (px + q)(rx + s)$

What is the axis of symmetry in a quadratic function?

The vertical line $x = -\frac{b}{2a}$, which passes through the vertex.

Completing the square

A method to rewrite a quadratic equation in vertex form by adding and subtracting the square of half the coefficient of $x$.

Definition of a function

A relation where each input has exactly one output.

Domain of a function

The set of all possible input values (x-values) for the function.

Range of a function

The set of all possible output values (y-values) of the function.

What is a linear function?

A function whose graph is a straight line, typically in the form $y = mx + b$.

Inverse of a function

A function that reverses the effect of the original function, such that if $f(x) = y$, then $f^{-1}(y) = x$.

What is the purpose of the substitution method?

To solve systems of equations by solving one equation for one variable and substituting the result into the other equation.

Intercepts of a graph

Points where the graph crosses the axes: $x$-intercept (set $y=0$), $y$-intercept (set $x=0$).

What transformation does $y = f(x) + c$ represent?

Vertical shift upward by $c$ units.

What transformation does $y = f(x - c)$ represent?

Horizontal shift to the right by $c$ units.

Absolute value function

$f(x) = |x|$

Piecewise function

A function defined by different expressions for different intervals of the domain.

How to find the zeros of a function?

Set the function equal to zero and solve for $x$.

How does $y = -f(x)$ affect a graph?

It reflects the graph of $f(x)$ across the x-axis.

What does it mean for functions to be inverses?

Two functions $f$ and $g$ are inverses if $f(g(x)) = x$ and $g(f(x)) = x$ for all $x$ in the domain of $g$ and $f$, respectively.

Exponential function form

$y = a \cdot b^x$

Logarithmic function

The inverse of an exponential function, $y = \log_b(x)$

Property of logarithms: Product rule

$\log_b(MN) = \log_b(M) + \log_b(N)$

Property of logarithms: Quotient rule

$\log_b(\frac{M}{N}) = \log_b(M) - \log_b(N)$

Property of logarithms: Power rule

$\log_b(M^p) = p \cdot \log_b(M)$

How do you solve an absolute value equation?

Set up two equations: one with the positive value and one with the negative value, then solve each.

What is a system of equations?

A set of two or more equations with the same variables.

How do you solve a system of equations by graphing?

Graph each equation and find the point(s) where the graphs intersect.

What is the solution of a system of linear inequalities?

The region of the graph where the shaded areas of the inequalities overlap.

What is the vertex of a parabola?

The highest or lowest point on the graph of a quadratic function.

What does $f(x + c)$ do to the graph of $f(x)$?

Shifts it horizontally to the left by $c$ units.

How is the graph of $f(x)$ affected by $cf(x)$, where $c > 1$?

The graph is vertically stretched.

Polynomial function

A function that can be expressed in the form $a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0$.

End behavior of polynomials

The direction the graph approaches as $x \to \pm\infty$, determined by the leading term.

What is the remainder theorem?

When a polynomial $f(x)$ is divided by $x - a$, the remainder is $f(a)$.