This video describes how the symmetry of quadratic functions can be used to make sense of the formula for the line of symmetry of a parabola.
CCSS.MATH.CONTENT.HSF.IF.C.8 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
CCSS.MATH.CONTENT.HSF.IF.C.8.A Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.
Weiss, M. (2016). Solving and graphing quadratics with symmetry and transformations. Mathematics Teacher, 110(5).
Symmetry is helpful when graphing. If a graph is symmetric and I know what half of it looks like, I already know what the other half looks like too! A great thing about quadratic functions is that they have symmetry. So our key question is, how can understanding symmetry help us graph quadratic equations?
First, it’s important for students to explore the symmetry of parabolas. You could ask students to graph several parabolas and then draw in the line of symmetry for each graph. Then, ask them to discuss what they notice about the line of symmetry and how it relates to the graph.
Some guiding questions are things like, do you notice anything special about where the line of symmetry intersects the parabola?
Does the line of symmetry change if you shift the graph up or down? This relates to changing the constant term of the equation. What about if I change the quadratic term? The linear term?
Another cool thing about this kind of symmetry is that whenever you have one point, you get another one for free. Like, if I know negative-two four is on a parabola, and the line of symmetry is x equals 0, then I know two four is another point on the parabola.
Line of Symmetry
Many textbooks give the formula for determining the line of symmetry from an equation in standard form and then have students use that formula to determine the vertex and sketch the graph. Most books don’t explain where in the world “x equals negative B over 2 A” comes from, but students can explain it by connecting to their prior knowledge of midpoints and linear equations.
To begin, ask students whether they can find the line of symmetry if given two points at the same height, such as the x-intercepts. Spoiler alert, they can! The line of symmetry passes through the midpoint.
So now that we know the x-intercepts are useful, students can work to find the intercepts for a quadratic equation of this form. It might go like this. Remember C doesn’t change the line of symmetry because it just shifts the graph up or down, so we can ignore it and just think about an equation of the form “y equals A x-squared plus B x”.
Students should recall that x-intercepts occur when y equals 0. So the x-intercepts are the solutions to this equation, so let’s factor it.
When is this satisfied? When x = 0, which gives us the x-intercept zero zero; or when ax + b = 0 which gives us the other x-intercept, negative B over A, zero.
So now that we have our two points with the same height we know our line of symmetry is right smack between them at “x equals negative b over two a”! The mystery of the line of symmetry formula has been solved!