How to Graph Quadratic Functions?
Graphing quadratic functions produces a distinctive U-shaped curve called a parabola. Whether you start from standard form, vertex form, or a table of values, the process always comes down to locating four key features: the vertex, the axis of symmetry, the x-intercepts (if any), and the y-intercept. This guide walks through each method with worked examples.
What Does the Graph of a Quadratic Function Look Like?
The graph of \(\color{blue}{f(x) = \text{ ax }^{2} + \text{ bx } + c}\) is a parabola — a smooth, symmetric curve. It opens upward when \(\color{blue}{a > 0}\) and downward when \(\color{blue}{a < 0}\). The narrower the parabola, the larger the absolute value of \(\color{blue}{a}\).
How to Graph a Quadratic Function
Step 1: Find the Vertex
The vertex is the turning point of the parabola. Use the formula:
\(\color{blue}{x = -b \div (2a)}\), then compute \(\color{blue}{y = f(x)}\).
Step 2: Find the Axis of Symmetry
The axis of symmetry is the vertical line through the vertex: \(\color{blue}{x = -\frac{b}{(2a)}}\).
Step 3: Find the y-Intercept
Substitute \(\color{blue}{x = 0}\): the y-intercept is \(\color{blue}{(0, c)}\).
Step 4: Find the x-Intercepts
Solve \(\color{blue}{\text{ ax }^{2} + \text{ bx } + c = 0}\) by factoring or the quadratic formula. There may be 0, 1, or 2 real roots.
Step 5: Plot and Connect
Plot the vertex, intercepts, and at least one additional point on each side of the axis of symmetry (using the table of values). Connect the points with a smooth curve.
Step-by-Step Summary
- Identify \(\color{blue}{a}\), \(\color{blue}{b}\), \(\color{blue}{c}\) and determine if the parabola opens up or down.
- Calculate the vertex: \(\color{blue}{x = -\frac{b}{(2a)}}\); find \(\color{blue}{y}\).
- Write the axis of symmetry: \(\color{blue}{x = -\frac{b}{(2a)}}\).
- Find the y-intercept: \(\color{blue}{(0, c)}\).
- Find x-intercepts if they exist.
- Build a small table of values around the vertex for precision.
- Plot all points and draw the parabola.
Watch: Graphing a Parabola Using Roots and Vertex
Khan Academy shows how to use the roots and vertex to sketch a parabola efficiently:
Graphing Quadratic Functions – Worked Examples
Example 1: Graph \(\color{blue}{f(x) = x^{2} – 2x – 3}\).
\(\color{blue}{a=1, b=-2, c=-3}\). Opens upward.
Vertex: \(\color{blue}{x = 1}\), \(\color{blue}{f(1) = 1 – 2 – 3 = -4}\) → vertex \(\color{blue}{(1, -4)}\).
y-intercept: \(\color{blue}{(0, -3)}\).
x-intercepts: \(\color{blue}{x^{2} – 2x – 3 = (x – 3)(x + 1) = 0}\) → \(\color{blue}{x = 3}\) and \(\color{blue}{x = -1}\).
Plot vertex, intercepts at \(\color{blue}{(-1, 0)}\), \(\color{blue}{(3, 0)}\), and \(\color{blue}{(0, -3)}\). Connect with a smooth curve.
Example 2: Graph \(\color{blue}{g(x) = -x^{2} + 4x}\).
\(\color{blue}{a=-1, b=4, c=0}\). Opens downward.
Vertex: \(\color{blue}{x = 2}\), \(\color{blue}{g(2) = -4 + 8 = 4}\) → vertex \(\color{blue}{(2, 4)}\).
y-intercept: \(\color{blue}{(0, 0)}\).
x-intercepts: \(\color{blue}{-x(x – 4) = 0}\) → \(\color{blue}{x = 0}\) and \(\color{blue}{x = 4}\).
The parabola arcs downward from vertex \(\color{blue}{(2, 4)}\) through both intercepts.
Example 3: Graph \(\color{blue}{h(x) = 2x^{2} + 4x + 2}\).
\(\color{blue}{a=2, b=4, c=2}\). Opens upward (narrow).
Vertex: \(\color{blue}{x = -1}\), \(\color{blue}{h(-1) = 2 – 4 + 2 = 0}\) → vertex \(\color{blue}{(-1, 0)}\).
y-intercept: \(\color{blue}{(0, 2)}\).
x-intercept (double root): \(\color{blue}{x = -1}\) only (touches but does not cross).
More Practice: Graphing a Parabola with a Table of Values
Khan Academy uses a value table to produce an accurate parabola step by step:
Exercises for Graphing Quadratic Functions
- Find the vertex and y-intercept of \(\color{blue}{f(x) = x^{2} + 4x + 3}\).
- Find the x-intercepts of \(\color{blue}{f(x) = x^{2} – 9}\).
- Does \(\color{blue}{f(x) = -2x^{2} + 8}\) open up or down? State the vertex.
- What is the axis of symmetry of \(\color{blue}{f(x) = 3x^{2} – 6x + 1}\)?
- Graph \(\color{blue}{f(x) = x^{2} – 4x + 4}\) (state vertex, axis, intercepts).
Answers
- Vertex: \(\color{blue}{(-2, -1)}\); y-intercept: \(\color{blue}{(0, 3)}\)
- \(\color{blue}{x = \pm 3}\)
- Opens downward; vertex: \(\color{blue}{(0, 8)}\)
- \(\color{blue}{x = 1}\)
- Vertex: \(\color{blue}{(2, 0)}\); axis: \(\color{blue}{x = 2}\); double root at \(\color{blue}{x = 2}\); y-intercept: \(\color{blue}{(0, 4)}\)
Free Graphing Quadratic Functions Worksheet
Ready to practice on your own? Download our free Graphing Quadratic Functions worksheet below, work through each problem at your own pace, and then check your answers. If a few give you trouble, scroll back up to the worked examples and try again — steady practice is the surest way to master Graphing Quadratic Functions before a quiz or test.
Download Graphing Quadratic Functions Worksheet
Frequently Asked Questions
What is the fastest way to find the vertex?
Use the formula \(\color{blue}{x = -\frac{b}{(2a)}}\) to get the x-coordinate, then substitute into the function to get y. This is faster than completing the square for most standard-form problems.
What if a quadratic has no x-intercepts?
Plot the vertex and y-intercept, then use a table of values for extra points. The parabola still has its characteristic shape — it just doesn’t cross the x-axis.
How does \(\color{blue}{a}\) affect the width of the parabola?
Larger \(\color{blue}{|a|}\) makes the parabola narrower (steeper); smaller \(\color{blue}{|a|}\) makes it wider. For example, \(\color{blue}{y = 5x^{2}}\) is much narrower than \(\color{blue}{y = 0.2x^{2}}\).
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