What the exam asks
| Typical stem | What you must do |
|---|---|
| Which of the following is an -intercept of the graph of ? | Set each factor equal to zero |
| The graph of is shown. Which of the following could define ? |
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Lesson 6 of 6 · Advanced math
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This topic collects the “function sense” questions in Advanced Math: reading zeros from factors, matching a polynomial to its graph, using the factor and remainder theorems, spotting where a rational function is undefined, shifting and reflecting graphs, and evaluating composite functions. The items are short, but they sit in the hard third of the module and hinge on one precise idea, so a single sign error costs the point.
| Typical stem | What you must do |
|---|---|
| Which of the following is an -intercept of the graph of ? | Set each factor equal to zero |
| The graph of is shown. Which of the following could define ? |
3 questions on polynomial, rational and transformed functions. Every option is explained once you answer.
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The first 3 of 10 cards for this topic. Sign in and finish the lesson to review them with spaced repetition.
Factor theorem
| Match zeros, multiplicity and end behavior |
| If is a factor of , what is the value of ? | Factor theorem: |
| When is divided by , the remainder is 7. Which must be true? | Remainder theorem: |
| For what value of is undefined? | Find the zeros of the original denominator |
| The graph of is the graph of shifted 3 units left. Which equation defines ? | Replace with |
| What is ? (formulas or a table) | Evaluate from the inside out |
For a polynomial and a number , these statements all say the same thing:
To get a zero from a factor, solve the factor, don’t just read it: gives , and gives .
| Feature | What the graph does |
|---|---|
| Factor to an odd power, such as | crosses the -axis |
| Factor to an even power, such as | touches the axis and turns back |
| Even degree, positive leading coefficient | both ends rise |
| Odd degree, positive leading coefficient | falls on the left, rises on the right |
| Negative leading coefficient | the picture is flipped over the -axis |
The sign of a polynomial changes only at zeros of odd multiplicity. That is what makes sign-chart questions quick.
When is divided by , the remainder is . So:
You never need long division on the SAT. Just substitute.
| Equation | Effect on the graph of | Effect on a point |
|---|---|---|
| up | ||
| right | ||
| left | ||
| reflect over the -axis | ||
| reflect over the -axis | ||
| vertical stretch by |
Changes outside act on in the direction you’d expect. Changes inside act on in the opposite direction. Vertical stretches and reflections never move -intercepts, because .
means “apply first, then ”. If and , then , but . The order matters.