Graphing Quadratic Functions
A visual guide to understanding parabolas
Graphing a quadratic function becomes much easier when you know what to look for first. A quadratic graph is called a parabola, and its shape, turning point, symmetry, and intercepts all come from the equation.
What Is a Parabola?
A parabola is the U-shaped or upside-down U-shaped graph of a quadratic function. It has a line of symmetry and one special turning point called the vertex.
Standard Form
The coefficient a cannot equal zero.
Step 1: Check the Direction
The sign of a tells you which way the parabola opens.
When a Is Positive
If a > 0, the parabola opens upward and the vertex is its minimum point.
f(x) = x2
When a Is Negative
If a < 0, the parabola opens downward and the vertex is its maximum point.
f(x) = −x2
Width of the Parabola
If |a| is greater than 1, the parabola is narrower than y = x2. If |a| is between 0 and 1, it is wider.
Step 2: Find the Axis of Symmetry
The axis of symmetry is the vertical line that divides the parabola into two matching halves.
Axis of Symmetry Formula
Worked Example
For f(x) = x2 − 6x + 5, a = 1 and b = −6.
x = −(−6) ÷ 2(1) = 3
The axis of symmetry is x = 3.
Step 3: Find the Vertex
After finding the x-coordinate of the vertex, substitute it into the original function to find the y-coordinate.
Using f(x) = x2 − 6x + 5 and x = 3:
f(3) = 32 − 6(3) + 5 = 9 − 18 + 5 = −4
The vertex is (3, −4). Because the parabola opens upward, this is the minimum point.
Step 4: Find the Y-Intercept
To find the y-intercept, set x equal to 0. In standard form, the y-intercept is always (0, c).
Example
For f(x) = 2x2 + 3x − 5:
f(0) = −5 → y-intercept = (0, −5)
Step 5: Find the X-Intercepts
The x-intercepts are the points where the parabola crosses the x-axis. At those points, the y-value is 0, so solve f(x) = 0.
Factored Example
For f(x) = (x − 2)(x + 1), set each factor equal to zero:
x − 2 = 0 → x = 2
x + 1 = 0 → x = −1
The x-intercepts are (2, 0) and (−1, 0).
Vertex Form Makes Graphing Faster
The vertex is (h, k).
For f(x) = −2(x + 3)2 + 4, the vertex is (−3, 4). The negative value of a means the graph opens downward, and the factor 2 makes it narrower.
A Reliable Graphing Checklist
1. Check whether the parabola opens up or down.
2. Find the axis of symmetry.
3. Find and plot the vertex.
4. Find the y-intercept and x-intercepts when possible.
5. Use symmetry to plot matching points on both sides.
Common Mistakes to Avoid
Sign error in the vertex: In x − h, the vertex x-coordinate is h. Therefore x + 3 means h = −3.
Forgetting the 2a: The axis formula is x = −b ÷ 2a, not x = −b ÷ 2.
Confusing intercepts: The y-intercept has x = 0; x-intercepts have y = 0.
Drawing without symmetry: Reflect points across the axis to keep the graph accurate.
Practice Problems
1. Does f(x) = −2x2 + 4x + 1 open upward or downward?
2. Find the axis of symmetry of f(x) = x2 − 10x + 7.
3. Find the vertex of f(x) = (x − 2)2 − 3.
4. Find the x-intercepts of f(x) = (x − 4)(x + 2).
Tap to reveal the answers
1. Down. 2. x = 5. 3. (2, −3). 4. (4, 0) and (−2, 0).
Final Takeaway
To graph a quadratic, begin with the sign of a, find the axis of symmetry and vertex, then plot the intercepts. Vertex form shows the graph’s location quickly, while standard form helps you identify the coefficients and y-intercept.
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