How to Graph a Quadratic and Find Intercepts, Vertex, & Axis of Symmetry!

Introduction

In this tutorial, you will learn how to graph a quadratic function and find key features such as the axis of symmetry, vertex, x-intercepts, and y-intercepts. Understanding these concepts is essential for analyzing parabolas in algebra and calculus. By the end, you will have the skills to graph quadratics without a graphing calculator.

Step 1: Understand What a Quadratic Function Is

A quadratic function is a polynomial function of degree two, typically written in the form:

[ f(x) = ax^2 + bx + c ]

Where:

• ( a ), ( b ), and ( c ) are constants,
• ( a \neq 0 ),
• The graph of a quadratic function is a parabola.

Practical Tip

• The value of ( a ) determines the direction of the parabola:
  • If ( a > 0 ), the parabola opens upwards.
  • If ( a < 0 ), it opens downwards.

Step 2: Identify the Vertex of the Parabola

The vertex is the highest or lowest point on the graph of the quadratic function. To find the vertex, use the formula:

[ x = -\frac{b}{2a} ]

Steps to Find the Vertex

1. Identify the values of ( a ) and ( b ) from your quadratic equation.
2. Substitute these values into the vertex formula to find ( x ).
3. Substitute the value of ( x ) back into the original equation to find the ( y )-coordinate of the vertex.

Example

For the function ( f(x) = 2x^2 + 4x + 1 ):

• ( a = 2 ), ( b = 4 )
• Calculate ( x = -\frac{4}{2 \times 2} = -1 )
• Find ( f(-1) = 2(-1)^2 + 4(-1) + 1 = -1 )
• Thus, the vertex is at (-1, -1).

Step 3: Find the Axis of Symmetry

The axis of symmetry is a vertical line that passes through the vertex. It can be found using the same ( x ) value obtained from the vertex calculation.

Axis of Symmetry Formula

[ x = -\frac{b}{2a} ]

Example

Using the previous example, the axis of symmetry is ( x = -1 ).

Step 4: Calculate the X-Intercepts

The x-intercepts are the points where the graph crosses the x-axis. To find them, set ( f(x) = 0 ) and solve for ( x ).

Steps to Find X-Intercepts

1. Set the quadratic equation to zero: [ ax^2 + bx + c = 0 ]
2. Use the quadratic formula: [ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ]
3. Calculate the two possible values for ( x ).

Example

For ( 2x^2 + 4x + 1 = 0 ):

• ( a = 2 ), ( b = 4 ), ( c = 1 )
• Calculate the discriminant: ( b^2 - 4ac = 4^2 - 4 \times 2 \times 1 = 0 )
• Since the discriminant is 0, there is one x-intercept at: [ x = \frac{-4}{2 \times 2} = -1 ]

Step 5: Determine the Y-Intercept

The y-intercept is found by evaluating the function at ( x = 0 ).

Steps to Find the Y-Intercept

1. Substitute ( x = 0 ) into the function: [ f(0) = c ]
2. The value of ( c ) gives you the y-intercept.

Example

For ( f(x) = 2x^2 + 4x + 1 ):

• The y-intercept is ( f(0) = 1 ), so the point is (0, 1).

Step 6: Graph the Quadratic Function

Now that you have found the vertex, axis of symmetry, x-intercepts, and y-intercept, you can graph the quadratic.

Steps to Graph

1. Plot the vertex on the graph.
2. Draw the axis of symmetry through the vertex.
3. Plot the x-intercept(s) and the y-intercept.
4. Sketch the parabola, ensuring it opens in the correct direction based on the value of ( a ).

Common Pitfalls

• Miscalculating the vertex or intercepts.
• Forgetting to check the sign of ( a ) to determine the direction of the parabola.

Conclusion

You have learned how to graph a quadratic function by finding its vertex, axis of symmetry, and intercepts. Practice these steps with different quadratic equations to reinforce your understanding. Next, try applying these concepts to real-world problems or explore more advanced topics like transformations of quadratic functions.