Algebra II

Topics

Absolute Value Functions & Equations — transformations of \(f(x)=a|x-h|+k\), and solving \(|x|=c\) and \(|ax+b|=c\).

Applications of Linear Relationships — systems, inequalities, and linear models drawn from data.

You'll be able to…

• Graph \(f(x)=a|x-h|+k\) and read its vertex, axis, and direction of opening.
• Solve absolute-value equations as two cases and identify when there is no solution.
• Solve and graph systems of linear equations and inequalities.
• Build a linear model from a context and interpret slope and intercept.

Worked example

Topics

Quadratic Functions & Equations — vertex and standard form, completing the square, and the quadratic formula.

Applications of Quadratics — modeling area, projectile motion, and optimization; interpreting the vertex as a maximum or minimum.

You'll be able to…

• Convert between standard form \(ax^2+bx+c\) and vertex form \(a(x-h)^2+k\).
• Solve quadratics by factoring, completing the square, and the quadratic formula.
• Use the discriminant \(b^2-4ac\) to predict the number of real roots.
• Model a projectile or optimization problem and interpret its vertex.

Worked example

Topics

Composing & Decomposing Functions — \((f\circ g)(x)\), breaking a function into a chain, and inverse relationships.

Attributes of Cubic Functions — end behavior, inflection, and the shape of \(f(x)=ax^3+\dots\).

Relating Factors & Zeros — the factor theorem and reading a polynomial's roots from its graph.

You'll be able to…

• Compose and decompose functions and evaluate \((f\circ g)(x)\).
• Describe the end behavior and key attributes of a cubic function.
• Connect factors, zeros, and \(x\)-intercepts using the factor theorem.
• Build a polynomial from its roots and sketch its graph.

Worked example

Topics

Rational Functions — \(f(x)=\dfrac{1}{x}\), vertical and horizontal asymptotes, and operations on rational expressions.

Radical Functions — \(f(x)=\sqrt[n]{x}\), domain restrictions, and transformations.

Radical Equations — solving \(\sqrt{x}=c\) and checking for extraneous roots.

You'll be able to…

• Graph rational functions and locate vertical and horizontal asymptotes.
• Add, subtract, multiply, and divide rational expressions.
• Graph radical functions and state their domain and range.
• Solve radical equations and identify extraneous solutions.

Worked example

Topics

Exponential & Logarithmic Functions — growth and decay \(f(x)=a\cdot b^{x}\), and the inverse relationship \(y=\log_b x\).

Exponential & Logarithmic Equations — solving with the laws of logarithms and applying them to models.

You'll be able to…

• Graph exponential and logarithmic functions and identify their asymptotes.
• Convert between exponential and logarithmic form: \(b^{y}=x \iff y=\log_b x\).
• Apply the laws of logarithms to expand, condense, and solve equations.
• Build and interpret exponential growth, decay, and compound-interest models.

Worked example