Parabola Features on the SAT
Axis, Vertex, and Intercepts
Build equations from context, spot patterns fast, and practice with intent.
Why the SAT Emphasizes Parabola Features
The SAT rarely asks you to graph an entire parabola from scratch. Instead, it asks for key features like the axis of symmetry, the vertex, or the intercepts. If you can find those quickly, you can answer most quadratic graph questions without guessing.
This lesson shows how to extract those features from standard form, how to interpret them in context, and how to avoid mixing up the $x$ value of the vertex with the maximum or minimum value of the function.
A Simple Definition Unlocks Parabola Features
In $y = ax^2 + bx + c$, the axis of symmetry is $x = - \frac{b}{2a}$. Plugging that $x$ value into the equation gives the vertex $y$ value.
Intercepts come from setting $y = 0$ for $x$ intercepts, or $x = 0$ for the $y$ intercept. Because parabolas are symmetric, the vertex sits exactly halfway between the $x$ intercepts when they exist.
Work Through Parabola Features Step by Step
For $y = -2x^2 + 4x + 1$, find the axis of symmetry (and vertex).
Find the axis and vertex for $y = -2x^2 + 4x + 1$.
Identify the coefficient $a$ from the quadratic.
Identify the coefficient $b$ from the quadratic.
Use the axis of symmetry formula
Substitute the values into the formula.
Simplify the expression to make the next step clear.
Plug $x = 1$ into the equation
Compute the vertex $y$ value
Use Desmos to Check Parabola Features
For $y = -2x^2 + 4x + 1$, find the axis of symmetry (and vertex).
Desmos shows the vertex and intercepts when you click the graph. This is extremely fast for feature questions.
y = -2x^2 + 4x + 1
Use algebra when the question asks for an exact expression. Use Desmos when you need a quick coordinate or a check.
Desmos is faster when you need to read the vertex or intercepts quickly from a graph. Algebra is faster when you need exact symbolic values.
Expert move: Graph the quadratic and click the $x$-intercepts and vertex to read solutions and key features; the graph makes it clear when there are 0, 1, or 2 real roots.
Precision check: Use Desmos for decimal answers or verification, but convert to a fraction if the choices are exact and apply any context restrictions.
- Desmos features used: vertex display, intercepts.
- Common mistake: using $\frac{b}{2a}$ instead of $-\frac{b}{2a}$.
Practice Parabola Features with SAT-Style Questions
Find axis, vertex, or intercepts.
For , what is the axis of symmetry?
If , does the parabola have a maximum or minimum?
What is the vertex of ?
For , what is the intercept?
Key Takeaways to Remember for Parabola Features
- Axis of symmetry is $x = -\frac{b}{2a}$.
- The sign of $a$ tells you if the vertex is a max or min.
- Desmos gives quick vertex and intercept checks.
