Digital SAT Math: Advanced Math

Advanced Math is the Digital SAT Math domain that rewards students who can think in functions, not just in numbers. It makes up a large share of the test, and it is built around algebra beyond linear equations: quadratic functions, polynomials, exponentials, and the language of function notation. If you are comfortable translating between an equation, a graph, and a verbal description, you will recognize most “hard” problems as familiar patterns.

This article breaks down what Advanced Math covers, what the test is really asking, and how to approach the most common question types efficiently.

What “Advanced Math” means on the Digital SAT

On the Digital SAT, Advanced Math focuses on:

• Quadratic functions and their features (vertex, zeros, maximum/minimum, symmetry)
• Polynomials (factoring, roots, multiplicity, end behavior)
• Exponential functions (growth and decay, interpretation of parameters)
• Function notation and function analysis (evaluating, interpreting, combining, transforming)
• Transformations of graphs and equations (shifts, stretches, reflections)

These topics are not isolated. A single question may combine function notation with a quadratic model, or ask you to compare an exponential expression to a real-world context. The skill is not memorizing tricks; it is recognizing structure.

Function notation and function thinking

Function notation shows up constantly because it forces precision. When you see $f(x)$, read it as “the output of the function when the input is $x$.”

Evaluating and interpreting $f(x)$

Many problems simply test whether you can substitute correctly:

• If $f(x)=2x^2-3x+1$, then $f(4)$ means plug in $x=4$.
• If a context says $f(t)$ is the height of a ball at time $t$, then $f(2)$ is the height at 2 seconds.

The higher-level version is interpretation: if $f(a)=0$, that means the input $a$ produces an output of zero. On a graph, that corresponds to an $x$-intercept.

Combining functions

You may be asked about expressions like:

• $f(x)+g(x)$
• $f(x)g(x)$
• $f(g(x))$ (composition)

Composition is the most error-prone. The key is order: $f(g(x))$ means apply $g$ first, then feed the result into $f$. A quick habit: rewrite $g(x)$ as a single “chunk” and substitute that chunk everywhere you see $x$ in $f$.

Quadratic functions: the SAT’s favorite non-linear model

Quadratics appear in multiple forms, and the Digital SAT expects you to move between them.

The three common forms

1. Standard form: $y=a{x}^2+bx+c$

Best for identifying the $y$-intercept ($c$) and setting up algebra.

1. Vertex form: $y=a(x-h{)}^2+k$

Best for reading the vertex $(h,k)$ and understanding transformations.

1. Factored form: $y=a(x-r_1)(x-r_2)$

Best for identifying zeros (roots) $r_1$ and $r_2$.

A typical question asks you to choose an equivalent form, interpret a feature, or determine how changing a parameter affects the graph.

Key features to know cold

• Vertex: For $y=a(x-h{)}^2+k$, vertex is $(h,k)$.

For standard form, the axis of symmetry is $x=-\frac{b}{2a}$, and the vertex occurs at that $x$-value.

• Maximum/minimum: If $a>0$, the parabola opens up and the vertex is a minimum. If $a<0$, it opens down and the vertex is a maximum.

• Zeros and factoring: Solving $a{x}^2+bx+c=0$ often means factoring when possible. When factoring is not clean, the SAT may instead provide information (like one root) or ask about the number of solutions.

Transformations of quadratics

Transformations are a major Advanced Math theme. If you start with $y=x^2$:

• $y=(x-3{)}^2$ shifts right 3
• $y=(x+3{)}^2$ shifts left 3
• $y=x^2+5$ shifts up 5
• $y=2x^2$ is a vertical stretch by 2 (narrower)
• $y=-x^2$ reflects over the $x$-axis

The SAT often disguises a transformation problem as an “equivalent equation” question. If you can recognize vertex form, you can answer quickly without expanding.

Polynomials: structure, roots, and behavior

Polynomial questions frequently test whether you understand how roots, factors, and graphs connect.

Factoring and solving

If a polynomial expression equals zero, the fastest method is usually factoring:

• Factor out a greatest common factor first.
• Recognize special products like $(x^2-9)=(x-3)(x+3)$.
• Look for common factor patterns like grouping.

A common trap: solving $p(x)=0$ is different from simplifying $p(x)$. Do not cancel factors across an equals sign unless you understand what happens to solutions.

Roots and multiplicity

If a polynomial has a factor $(x-2{)}^2$, then $x=2$ is a root with multiplicity 2. Graphically, that root typically touches the $x$-axis and turns around instead of crossing.

Even if the SAT does not use the term “multiplicity,” it may describe the behavior: “the graph touches the $x$-axis at $x=2$,” which hints at an even power.

End behavior and degree

For a polynomial $p(x)$:

• The highest power (degree) dominates for large $|x|$.
• The sign of the leading coefficient determines whether the graph rises or falls on the ends.

This helps in questions that ask you to choose a graph that matches an equation, or predict what happens as $x$ increases.

Exponential functions: growth, decay, and interpretation

Exponential functions appear in both pure algebra and real-world modeling. The test cares about the meaning of parameters as much as computation.

A common structure is:

$$y=a\cdot b^x$$

• $a$ is the starting value when $x=0$
• $b$ is the growth factor per 1 unit increase in $x$
• If $b>1$, it is growth
• If $0<b<1$, it is decay

Reading an exponential model in context

If a quantity “increases by 20% each year,” the factor is $1.20$. If it “decreases by 15% each year,” the factor is $0.85$.

SAT questions may also present exponentials as repeated multiplication rather than percent language, so practice translating between the two. Also be ready to compare two exponentials by reasoning about their factors rather than expanding large numbers.

Typical Digital SAT Advanced Math question styles

Equivalent expressions and forms

You may be asked which expression is equivalent to a given function, or which form reveals a feature (vertex, roots, intercepts). The efficient approach is to choose the form that matches the question:

• Need zeros? Use factored form.
• Need vertex? Use vertex form.
• Need intercept? Use standard form.

Function analysis from a graph

Graph-based questions often ask for:

• The value of $f(c)$
• The solutions to $f(x)=0$
• A transformation that maps one graph to another

Treat the graph as a data source. Read coordinates carefully and remember that the Digital SAT interface can make small misreads costly.

Transformations and parameter changes

Expect questions like: “If $g(x)=f(x-4)+1$, how does the graph of $g$ compare to the graph of $f$?” Translate step by step:

• Inside change ($x-4$) affects horizontal shift
• Outside change ($+1$) affects vertical shift

Practical study priorities

1. Master function notation basics. Many missed problems come from simple substitution or misreading composition.
2. Get fluent in quadratic forms. Being able to switch between standard, vertex, and factored form saves time.
3. Practice transformations until they are automatic. Shifts and reflections are common and fast points when you recognize them.
4. Connect algebra to graphs. Roots, intercepts, and vertex are not separate facts; they are the same ideas in different representations.
5. Interpret exponential parameters. Know what $a$ and $b$ mean and how percent change becomes a factor.

Advanced Math on the Digital SAT is less about exotic algebra and more about disciplined interpretation. When you consistently ask, “What is the function doing, and what feature is the question targeting?” you turn complex-looking problems into straightforward decisions.