Template for graphing a parabola from a quadratic equation in standard form:
y  ax 2  bx  c
INSTRUCTIONS: This first page describes the steps you should go through in order to
graph a quadratic equation or inequality. The next page is an example of graphing a
quadratic by using these steps. The third page is a template you can print and complete
for each quadratic you need to graph.
STEP 1: Find the axis of symmetry.
The axis of symmetry is determined by the formula x 
b
2a
STEP 2: Find the y-value of the vertex.
To find the y-value of the vertex, substitute the x-value (from the axis of
symmetry in Step 1) back into the equation and solve for y.
STEP 3: Is this an upward- or downward-facing parabola?
To determine the orientation of the parabola, look at the coefficient of the xsquared term in the equation (the a-value). If a is negative, the parabola faces
down. If a is positive, the parabola faces up.
STEP 4: What two values will you choose to the left of the vertex?
Pick any two numbers smaller than the x-value of the vertex (from Step 1). Pick
numbers that are small in absolute value and that are easy to substitute in and
calculate with.
STEP 5: Find the y-values for those two points left of the vertex.
Substitute each of the x-values from Step 4, and find the associated y-values
STEP 6: What two values will you choose to the right of the vertex?
Pick any two numbers larger than the x-value of the vertex (from Step 1). Pick
numbers that are small in absolute value and that are easy to substitute in and
calculate with.
STEP 7: Find the y-values for those two points right of the vertex.
Substitute each of the x-values from Step 6, and find the associated y-values
STEP 8: Graph the quadratic equation or inequality
You now have the axis of symmetry from Step 1. Sketch in that line
You now have the y-value of the vertex from Step 2. Plot that point
You know whether the parabola faces up or down from Step 3.
You have two points on the graph from Step 4 and Step 5. Plot those points.
You have two points on the graph from Step 6 and Step 7. Plot those points.
“Connect the dots” and draw the graph of the parabola.
GRAPHING A QUADRATIC EQUATION – A SAMPLE PROBLEM:
Graph
y   x2  2x  6
STEP 1: Find the axis of symmetry. In this equation, a = -1 and b = 1, so the axis of symmetry is at
x
b (2) 2


1
2a 2(1) 2
STEP 2: Find the y-value of the vertex. Substitute in the x-value from step 1 and solve for y:
y   x 2  x  6   12  3 •1  6   1  3  6  5
STEP 3: Is this an upward- or downward-facing parabola? This faces down because of the negative sign
before the x2
STEP 4: What two values will you choose left of the vertex? Two numbers less than 1 are 0 and -1
STEP 5: Find the y-values for those two points left of the vertex.
y   x2  2x  6
y   x2  2 x  6
y  (0) 2  2(0)  6
y  (1) 2  2(1)  6
y  6 when x = 0
y  9 when x = -1
STEP 6: What two values will you choose right of the vertex? Two numbers larger than 1 are 2 and 3
STEP 7: Find the y-values for those two points right of the vertex.
y   x2  2x  6
y   x2  2 x  6
y  (2) 2  2(2)  6
y  (3) 2  2(3)  6
y  6 when x = 2
STEP 8: Graph the quadratic equation or inequality
y  9 when x = 3
USE THIS TEMPLATE TO GRAPH QUADRATIC EQUALITIES & INEQUALITIES
__________STEP 1: Find the axis of symmetry.
The axis of symmetry is determined by the formula x 
b
2a
__________STEP 2: Find the y-value of the vertex.
To find the y-value of the vertex, substitute the x-value (from the axis of symmetry in Step
1) back into the equation and solve for y.
__________STEP 3: Is this an upward- or downward-facing parabola?
To determine the orientation of the parabola, look at the coefficient of the x-squared term
in the equation (the a-value). If a is negative, the parabola faces down. If a is positive, the
parabola faces up.
__________STEP 4: What two values will you choose to the left of the vertex?
Pick any two numbers smaller than the x-value of the vertex (from Step 1). Pick numbers
that are small in absolute value and that are easy to substitute in and calculate with.
__________STEP 5: Find the y-values for those two points left of the vertex.
Substitute each of the x-values from Step 4, and find the associated y-values
__________STEP 6: What two values will you choose to the right of the vertex?
Pick any two numbers larger than the x-value of the vertex (from Step 1). Pick numbers
that are small in absolute value and that are easy to substitute in and calculate with.
__________STEP 7: Find the y-values for those two points right of the vertex.
Substitute each of the x-values from Step 6, and find the associated y-values
STEP 8: Graph the quadratic equation or inequality