Finding the Axis of Symmetry and the Vertex
Enter the equation for y = x2 into your calculator, and look at a table of values.
You should not be surprised to observe that when the values of x are negatives of one another, the y values are the same. The impact of this simple fact is that the vertical line passing through the vertex serves as a line of symmetry for the parabola. Take a piece of graph paper, plot these points, and graph the parabola. Fold the paper along the y-axis, and observe that when you do so, the two arcs of the parabola lie one on top of the other. The vertex of the parabola is the only point on the graph that does not have a mirror image.
Where is the axis of symmetry when the graph is moved vertically? It is still the y-axis. Where is the axis of symmetry when the graph is moved horizontally? It makes sense that the axis of symmetry also moves horizontally, but finding its equation can save you a lot of work. Take a look at the table of values for the function y = x2 − 2x. For what value of x do you see the same type of symmetry you saw with y = x2?
The y values are symmetric about x = 1. Repeat this process for y = x2 − 4x, y = x2 + 2x, and y = x2 + 4x. The axes of symmetry are at x = 2, x = −1, and x = −2, respectively. Does this help you see that when the equation of the parabola is y = ax2 + bx + c, the equation for the axis of symmetry will be x = -b/2a?
Writing the equation for the axis of symmetry is often a place where people lose points on tests, because they fail to write x = as part of their response.
Given the equation y = x2 − 3x, the formula states that the axis of symmetry should have the equation x = 3/2. Check to see that this agrees with your graph and its table of values.
Determine the equation for the axis of symmetry for the graph of y = 3x2 + 12x − 1.
a = 3, b = 12. The equation for the axis of symmetry is x = −12/2(3), or x = −2.
The vertex of the parabola is the point where the axis of symmetry intersects the parabola. Substitute the value for the axis of symmetry into the equation to find the coordinates of the vertex.
Determine the coordinates of the vertex of the parabola with equation y = 3x2 + 12x − 1.
The axis of symmetry is at x = −2. The vertex is at y = 3(-2)2 + 12(-2) − 1 = −13. The coordinates of the vertex are (-2, −13).
Find the equation for the axis of symmetry and the coordinates of the vertex of the parabola with equation y = −1/3 x2 + 2x + 2.
a = −1/3 and b = 2. The equation for the axis of symmetry isx = −2/2(-1/3) = 3. The y-coordinate for the vertex isy = −1/3(3)2 + 2(3) + 2 = 5. The coordinates of the vertex are (3, 5).