SAT — Polynomial Graphs

Chapter Overview & Weightage

Polynomial graphs (mostly quadratics, occasional cubics) appear 3-5 times on the Digital SAT Math section. Topics include: identifying zeros from a graph, recognising vertex form, predicting end behaviour, and translating algebraic features to graphical ones.

Test	Questions
Digital SAT	3-5
PSAT	2-4

Quadratics dominate. Cubic and quartic graphs appear less often but easy to recognise once you know the shapes.

Key Concepts You Must Know

Quadratic graph (parabola): opens up if leading coefficient $> 0$ , down if $< 0$ .
Vertex form: $y = a(x - h)^2 + k$ , vertex at $(h, k)$ .
Standard form: $y = ax^2 + bx + c$ , vertex $x$ -coord $= -b/(2a)$ .
Factored form: $y = a(x - r_1)(x - r_2)$ , zeros at $r_1, r_2$ .
Discriminant: tells how many real zeros (i.e. how many times graph crosses $x$ -axis).
End behaviour: degree and leading coefficient determine far-left and far-right.
Cubic graphs: at most 2 turning points; degree-3 polynomials.

Important Strategies

Form	Equation	What it shows
Standard	$ax^2 + bx + c$	$y$ -intercept = $c$
Vertex	$a(x - h)^2 + k$	Vertex $= (h, k)$
Factored	$a(x - r_1)(x - r_2)$	Roots $r_1, r_2$

Convert between forms using completing the square or factoring.

Degree even, leading coeff $> 0$ : both ends rise. Degree even, leading coeff $< 0$ : both ends fall. Degree odd, leading coeff $> 0$ : rises to right. Degree odd, leading coeff $< 0$ : rises to left.

Solved Sample Questions

Sample 1 (Digital SAT 2024)

The graph of $y = -2(x - 3)^2 + 5$ has its vertex at which point?

Vertex form gives $(h, k) = (3, 5)$ directly. The negative coefficient means the parabola opens downward, and $5$ is the maximum value.

Vertex: $(3, 5)$ .

Sample 2 (Digital SAT 2023)

The graph of $y = (x - 1)(x + 4)$ crosses the $x$ -axis at which points?

Setting $y = 0$ : $x = 1$ or $x = -4$ .

Crosses at $x = 1$ and $x = -4$ .

Sample 3 (PSAT)

Which equation could represent the graph that opens downward and has $x$ -intercepts at $-2$ and $5$ ?

Negative leading coefficient + roots at $-2, 5$ gives the factored form $y = -k(x + 2)(x - 5)$ for any $k > 0$ .

If we choose $k = 1$ : $y = -(x + 2)(x - 5) = -(x^2 - 3x - 10) = -x^2 + 3x + 10$ .

Difficulty Distribution

Sub-topic	Easy	Medium	Hard
Reading graph features	70%	25%	5%
Form conversions	30%	50%	20%
Cubics	40%	50%	10%
End behaviour	50%	40%	10%

The hardest questions ask you to convert between forms (vertex to factored, etc.) under time pressure.

Expert Strategy

Memorise the three forms and what each reveals. Match the question’s needed information to the right form — saves a lot of algebra.

For “which graph matches the equation” questions, find the $y$ -intercept ( $c$ ) and a couple of zeros. Three points is enough to pick the answer.

If the question gives you the vertex and asks for the equation, use vertex form. If it gives you the roots, use factored form. Don’t expand unnecessarily.

Common Traps

Confusing the sign of $h$ in vertex form. $y = a(x - h)^2 + k$ has vertex at $(h, k)$ , so $y = a(x + 3)^2$ has vertex at $(-3, 0)$ , not $(3, 0)$ .

Forgetting that the $y$ -intercept is $c$ in standard form, but $a \cdot r_1 \cdot r_2$ in factored form. They are equal, but easy to compute incorrectly under stress.

Misreading end behaviour for cubics. Odd-degree functions go in opposite directions at $\pm\infty$ , not the same.