Functions and their graphs
Students start the year looking closely at different kinds of math relationships and what their graphs look like. They learn to spot patterns, find key points, and describe how one quantity changes as another one shifts.
What does a student learn in
This is the year math stretches beyond straight lines into the curves and waves that describe the real world. Students study new families of functions, including polynomials, exponentials, logarithms, and the sine waves that repeat like tides or seasons. They also learn to draw careful conclusions about a whole group from a small sample. By spring, students can sketch a curve from its equation and explain what it predicts.
- Polynomial functions
- Exponential and logarithmic functions
- Trigonometry
- Rational expressions
- Statistical inference
- Function graphs
Year at a glance
How the year usually goes. Every school and district set their own curriculum, so treat this as a guide, not official pacing.
- 1
- 2
Polynomials and rational expressions
Students work with longer algebraic expressions and learn how to add, subtract, multiply, and divide them. They also solve equations that include fractions with variables, which shows up later in science and finance.
- 3
Exponential and logarithmic models
Students study growth and decay, the kind of math behind interest on a savings account or a population over time. They learn what a logarithm is and how to use one to undo an exponent.
- 4
Trigonometry and periodic patterns
Students extend trigonometry beyond triangles and use it to describe things that repeat, like tides, sound waves, or hours of daylight. They graph these wave patterns and use identities to simplify expressions.
- 5
Statistics and inference
Students close the year by using samples to make claims about larger groups, the kind of thinking behind polls and medical studies. They learn what makes a sample trustworthy and how confident a conclusion really is.
Mastery Learning Standards
The required skills a student should display by the end of Grade 12.
Standards for Mathematical Practice
Standards for Mathematical Practice
- Algebra II
Make Sense of Problems
Students read a math problem carefully, figure out what it's actually asking, and keep working even when the path forward isn't obvious. Getting unstuck is part of the work.
- Algebra II
Reason Abstractly
Students take a real situation, turn it into an equation or expression to solve it, then translate the answer back into plain language that makes sense in the original context.
- Algebra II
Construct Arguments
Students build a mathematical argument to support their answer, then explain where another student's reasoning goes wrong or how it holds up.
- Algebra II
Model with Mathematics
Students take a real situation, like figuring out loan payments or predicting ticket sales, and write an equation or draw a graph that helps make sense of it.
- Algebra II
Use Tools Strategically
Students choose the right tool for the math in front of them: a calculator, a quick estimate, or pencil-and-paper work. The skill is knowing which one fits the problem, not just reaching for the same tool every time.
- Algebra II
Attend to Precision
Students choose words, labels, and units carefully when solving problems. A missing negative sign or the wrong label can change the answer entirely.
- Algebra II
Use Structure
Students notice patterns and hidden structure in math problems, like recognizing that a complicated expression is really just a familiar form in disguise. That recognition helps them choose a faster, cleaner path to the answer.
- Algebra II
Express Regularity
Students notice when the same steps keep appearing in a problem and use that pattern as a shortcut or rule. Instead of redoing the work each time, they describe what's always true.
| Standard | Definition | Code |
|---|---|---|
| Make Sense of Problems Algebra II | Students read a math problem carefully, figure out what it's actually asking, and keep working even when the path forward isn't obvious. Getting unstuck is part of the work. | CT-MATH.MP.hs-algebra-2.1 |
| Reason Abstractly Algebra II | Students take a real situation, turn it into an equation or expression to solve it, then translate the answer back into plain language that makes sense in the original context. | CT-MATH.MP.hs-algebra-2.2 |
| Construct Arguments Algebra II | Students build a mathematical argument to support their answer, then explain where another student's reasoning goes wrong or how it holds up. | CT-MATH.MP.hs-algebra-2.3 |
| Model with Mathematics Algebra II | Students take a real situation, like figuring out loan payments or predicting ticket sales, and write an equation or draw a graph that helps make sense of it. | CT-MATH.MP.hs-algebra-2.4 |
| Use Tools Strategically Algebra II | Students choose the right tool for the math in front of them: a calculator, a quick estimate, or pencil-and-paper work. The skill is knowing which one fits the problem, not just reaching for the same tool every time. | CT-MATH.MP.hs-algebra-2.5 |
| Attend to Precision Algebra II | Students choose words, labels, and units carefully when solving problems. A missing negative sign or the wrong label can change the answer entirely. | CT-MATH.MP.hs-algebra-2.6 |
| Use Structure Algebra II | Students notice patterns and hidden structure in math problems, like recognizing that a complicated expression is really just a familiar form in disguise. That recognition helps them choose a faster, cleaner path to the answer. | CT-MATH.MP.hs-algebra-2.7 |
| Express Regularity Algebra II | Students notice when the same steps keep appearing in a problem and use that pattern as a shortcut or rule. Instead of redoing the work each time, they describe what's always true. | CT-MATH.MP.hs-algebra-2.8 |
Algebra II
Algebra II
- Algebra II
Functions and Graphs
Students read graphs of advanced functions, spotting where they rise, fall, level off, or repeat. The function types include polynomials, rationals, exponentials, logarithms, and trig curves.
- Algebra II
Polynomial and Rational
Students add, subtract, multiply, and divide expressions with variables and fractions that contain variables. They also solve equations built from those expressions.
- Algebra II
Trigonometry
Students use sine, cosine, and related functions to describe real-world patterns that repeat on a cycle, like sound waves, tides, or seasonal temperature changes. They also use trig identities to simplify and solve those models.
- Algebra II
Statistics and Probability
Students use data collected from a sample group to draw conclusions about a larger population. They apply statistical reasoning to decide what the sample likely tells us about the whole group.
| Standard | Definition | Code |
|---|---|---|
| Functions and Graphs Algebra II | Students read graphs of advanced functions, spotting where they rise, fall, level off, or repeat. The function types include polynomials, rationals, exponentials, logarithms, and trig curves. | CT-MATH.A2.hs-algebra-2.1 |
| Polynomial and Rational Algebra II | Students add, subtract, multiply, and divide expressions with variables and fractions that contain variables. They also solve equations built from those expressions. | CT-MATH.A2.hs-algebra-2.2 |
| Trigonometry Algebra II | Students use sine, cosine, and related functions to describe real-world patterns that repeat on a cycle, like sound waves, tides, or seasonal temperature changes. They also use trig identities to simplify and solve those models. | CT-MATH.A2.hs-algebra-2.3 |
| Statistics and Probability Algebra II | Students use data collected from a sample group to draw conclusions about a larger population. They apply statistical reasoning to decide what the sample likely tells us about the whole group. | CT-MATH.A2.hs-algebra-2.4 |
Assessments
The state tests students at this grade and subject take.
NAEP (National Assessment of Educational Progress)
Federally administered sample-based assessment in reading, mathematics, science, and writing. NAEP results inform state-by-state comparisons rather than individual student or school accountability.
- When given:
- biennial in winter
- Frequency:
- every two years
Common Questions
What math will students learn this year?
Students study a wider set of functions than before, including curves with multiple bends, fractions with variables, growth and decay patterns, and the wave shapes used for sound and seasons. They also learn how to draw conclusions about a large group from a small sample.
How can a parent help with homework at home?
Ask students to explain what the problem is asking before they pick up a pencil. If they get stuck, have them sketch a quick graph or try a smaller version of the problem first. Most of the work is about noticing patterns, so questions like what changes and what stays the same go a long way.
Why is this year harder than the first algebra class?
The functions get more complicated and the problems combine ideas from earlier years. Students also have to switch between an equation, a graph, and a table for the same situation. Expect more time per problem and more written reasoning.
How should the year be sequenced?
A common path starts with polynomials and rational expressions, moves into exponential and logarithmic functions, then trigonometry and periodic models, and ends with statistical inference from samples. Function analysis runs through every unit, so graphs and structure should come up daily.
Which topics usually need the most reteaching?
Logarithms, rational expressions, and the unit circle tend to be the stickiest. Students often need a refresh on factoring and on solving equations that produce extra answers that do not actually work. Plan for short review loops rather than one long fix later.
What does a calculator do and not do in this course?
A graphing tool helps students see the shape of a function and check answers, but it will not explain why a graph behaves the way it does. Students should still be able to sketch a basic curve by hand and read key features like intercepts and turning points without help.
How does the statistics unit fit with the rest of the year?
The statistics work is about using a sample to say something reasonable about a larger group, with attention to how the sample was collected. It connects to the rest of the year through careful reasoning and precise language, not through new algebra. Many teachers save it for the final stretch.
How do families know students are ready for the next math course?
By spring, students should be able to look at an unfamiliar function and describe its shape, find where it crosses the axes, and explain what the numbers mean in a real situation. They should also be able to solve an equation and check whether the answer actually fits the problem.
What home practice is worth ten minutes a day?
Reviewing one or two problems from class out loud is more useful than a worksheet. Ask students to talk through why each step works, not just what the answer is. Catching a small misunderstanding tonight saves an hour of confusion next week.