Engaging students: Finding the focus and directrix of a parabola

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Brittnee Lein. Her topic, from Precalculus: finding the focus and directrix of a parabola. What are the contributions of various cultures to this topic?

Parabolas (as we know them) were first written about in Apollonius’s Conics. Apollonius stated that parabolas were the result of a plane cutting a double right circular cone at an angle parallel to the vertical angle (α). So, what does that actually mean?

Well, if we take a vertical line and intersect it with a straight line at a fixed point, and then rotate that straight line around the fixed point we form the shape below: If the plane slices the cone at the angle β and β=α, a parabola is formed. This is still how we define parabolas today although you may not think about it that way. When you think of a parabola, you think of the equation $y = ax^2 +bx + c$. This equation is derived using the focus and the directrix. This video shows how to do so:

Understanding how the focus and directrix affect the equation of a parabola is crucial to understanding what each word means. According to mathwords.com, “For a given point, called the focus, and a given line not through the focus, called the directrix, a parabola is the locus of points such that the distance to the focus equals the distance to the directrix.” The directrix is a line perpendicular to the axis of symmetry and the focus falls on the line of the axis of symmetry. How can technology be used to effectively engage students with this topic?

This desmos activity can be used to show students how changing the focus, directrix, and vertex of the parabola affects the graph. https://www.desmos.com/calculator/y90ffrzmco

From this, students can shift values of the vertex and see that the directrix stays constant when the x-value is changed and that the focus remains constant when the y-value is shifted. If students change the value of the focus, they can see how it stretches and contracts the width of the parabola and how the directrix shifts. They can also see that when the focus is negative, the parabola opens downward and the directrix is positive. This website: https://www.intmath.com/plane-analytic-geometry/parabola-interactive.php Is also very helpful in showing the relationships between the focus, directrix and the graph of the parabolas because students can clearly see that the distance between a point on the parabola and the focus and the distance between that same point and the directrix are equal. What interesting (i.e., uncontrived) word problems using this topic can your students do now?

The website http://www.purplemath.com/modules/parabola4.htm has a lot of great real-world word problems involving finding the focus and the directrix of a parabola. For example, one of the questions is:  (This is a graph I made using desmos to model the situation at hand)

This problem requires a lot of prior knowledge of parabolas and really tests students’ ability to interpret information. From the question alone, the students can find the x-intercepts (-15,0) and (15,0) from the information “the base has a width of 30 feet”. They are also able to infer that the slope of the parabola will be negative because of the shape of an arch. The student must also know how to find the slope of the parabola using the x-intercepts, solving for the equation of the parabola using the x-intercepts and vertex and the equations for finding the focus and directrix from the given information. There are a few problems as involved as this one on the listed website above.