PSAT/NMSQT Math: Graphing Quadratics (Vertex, Axis of Symmetry, Intercepts)
A quadratic's graph is a parabola, and once you can identify its vertex, axis of symmetry, direction, and intercepts, you can sketch or interpret it without plotting a single extra point. This free, complete lesson covers reading the vertex from vertex form, finding the vertex and axis of symmetry from standard form, determining a parabola's direction and width, finding x-intercepts, and writing a quadratic's equation from its vertex or graph. Each idea is followed by a fully worked example, and then a set of original practice problems with instant feedback and full explanations. Everything here is free, and you can keep practicing afterward with the linked quizzes below.
1. The Vertex from Vertex Form
A quadratic written in vertex form, y = a(x − h)² + k, immediately reveals its vertex as the point (h, k). Be careful with signs: the h-value inside the parentheses is the opposite of what's written, since the form subtracts h.
Worked Example: Find the vertex of y = (x − 3)² + 2.
Comparing to y = a(x − h)² + k: h = 3, k = 2, so the vertex is (3, 2)
1. Find the vertex of y = (x + 4)² − 7.
Explanation: Rewrite as (x − (−4))² − 7, so h = −4 and k = −7, giving vertex (−4, −7).
2. Find the vertex of y − 6 = −2(x + 3)².
Explanation: Rewrite as y = −2(x + 3)² + 6, so h = −3 and k = 6, giving vertex (−3, 6).
3. Which quadratic has its vertex on the y-axis and passes through (1, 3)?
Explanation: A vertex on the y-axis means h = 0, matching y = 3x² (vertex at the origin). Checking (1, 3): 3(1)² = 3, which matches.
2. Vertex and Axis of Symmetry from Standard Form
A quadratic in standard form, y = ax² + bx + c, has axis of symmetry x = −b/(2a). This x-value is also the x-coordinate of the vertex; substitute it back into the original equation to find the y-coordinate.
Worked Example: Find the axis of symmetry and vertex of y = x² − 4x + 1.
Axis of symmetry: x = −(−4)/(2 · 1) = 2.
y = (2)² − 4(2) + 1 = 4 − 8 + 1 = −3, so the vertex is (2, −3)
1. Find the axis of symmetry of y = x² − 4x + 1.
Explanation: x = −b/(2a) = −(−4)/(2 · 1) = 4/2 = 2.
2. Find the vertex of y = −x² + 6x − 5.
Explanation: x = −6/(2 · −1) = 3. Then y = −(3)² + 6(3) − 5 = −9 + 18 − 5 = 4, so the vertex is (3, 4).
3. Determine the parabola's equation given a vertex at (1, −2) and passing through (3, 6), in vertex form y = a(x − 1)² − 2.
Explanation: Substitute (3, 6): 6 = a(3 − 1)² − 2 → 8 = 4a → a = 2, giving y = 2(x − 1)² − 2.
3. Direction, Width, and X-Intercepts
In y = ax² + bx + c (or a(x − h)² + k), the sign of a determines direction: positive a opens the parabola upward (with a minimum at the vertex), negative a opens it downward (with a maximum). A larger |a| makes the parabola narrower; a smaller |a| makes it wider. The x-intercepts are found by setting y = 0 and solving.
Worked Example: Find the x-intercepts of y = x² − 9.
Set y = 0: x² − 9 = 0 → x² = 9.
x = 3 or x = −3, so the x-intercepts are (3, 0) and (−3, 0)
1. Does y = −2(x + 1)² + 5 open upward or downward, and is it narrower or wider than y = x²?
Explanation: The negative coefficient (a = −2) means it opens downward, and |a| = 2 > 1 means it is narrower than y = x² (where a = 1).
2. Find the x-intercepts of y = x² − 9. (worked example above)
Explanation: x² = 9 gives x = ±3, so the x-intercepts are (3, 0) and (−3, 0).
3. A parabola shifts down 3 units from y = (x − 2)². Which equation represents the new parabola?
Explanation: Shifting a graph down means subtracting from the entire function's output (the k value), giving y = (x − 2)² − 3.
4. Writing Quadratic Equations from a Vertex or Graph
To write a quadratic's equation from its vertex, start with vertex form y = a(x − h)² + k using the known vertex (h, k), then substitute any other known point to solve for a. If a graph directly shows the vertex, simply read its coordinates off the graph.
Worked Example: A parabola has vertex (0, −4) and passes through (2, 0). Find its equation.
Start with y = a(x − 0)² − 4 = ax² − 4. Substitute (2, 0):
0 = a(2)² − 4 → 4a = 4 → a = 1, so y = x² − 4
1. Which equation matches a parabola with vertex (4, −3)?
Explanation: Substituting h = 4 and k = −3 into y = a(x − h)² + k gives y = a(x − 4)² − 3.
2. A graph shows a parabola with vertex at (−2, 5). What are the values of h and k in vertex form?
Explanation: The vertex (h, k) is read directly from the graph: h = −2, k = 5.
3. Which equation matches a parabola with vertex (0, −4) that passes through (2, 0)? (see worked example above)
Explanation: As shown in the worked example, solving for a using the given point gives a = 1, so y = x² − 4.
Common Mistakes to Avoid
- Reading the wrong sign for h in vertex form. Since the form subtracts h, y = (x + 3)² has h = −3, not h = 3.
- Forgetting to substitute the axis of symmetry back in to find the y-coordinate of the vertex. x = −b/(2a) only gives the x-coordinate; you still need to plug it back in for y.
- Mixing up which sign of a means "upward" versus "downward." Positive a opens upward (minimum); negative a opens downward (maximum).
- Assuming every quadratic has two real x-intercepts. Some parabolas touch the x-axis at exactly one point (a repeated root), and others never cross it at all.
- Solving for a using the vertex point itself instead of a different known point. Substituting the vertex into vertex form just confirms 0 = 0; you need a separate point to actually solve for a.
Frequently Asked Questions
What's the fastest way to find a parabola's vertex from standard form?
Use x = −b/(2a) to get the x-coordinate of the vertex, then substitute that value back into the original equation to find the y-coordinate. This is faster than completing the square for most PSAT problems.
How can I tell how wide or narrow a parabola is just from its equation?
Compare the absolute value of the leading coefficient a to 1. If |a| > 1, the parabola is narrower than y = x²; if |a| < 1 (a fraction), it's wider.
Where can I practice more problems like these?
The Graphing Quadratics quizzes in the PSAT/NMSQT Math Question Bank include additional original problems on this topic, along with quizzes covering every other Heart of Algebra, Advanced Math, Problem Solving and Data Analysis, and Geometry and Trigonometry skill tested on the PSAT/NMSQT. You can also browse the full PSAT/NMSQT Math Question Bank.