6.2: The Vertex of a Parabola

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Objectives:

  • Learn about the vertex form of a quadratic function
  • Find a formula given the vertex and another point on a parabola
  • Complete the square


Important Items

Definitions:

vertex, vertex form, axis of symmetry

Notes:

Expect students to be much less familiar with vertex form and completing the square than they were with standard form, factoring, and the quadratic formula. You will need to review completing the square with them.

Lesson Guide

Warm-Up

Have students do Problems 1 and 2.


Learn about the vertex form of a quadratic function

Review standard form and factored form from \S6.1. Discuss what information is highlighted in each form, i.e., in standard form we can clearly see the $y$-intercept and in factored form we can clearly see the $x$-intercepts. Introduce vertex form by sketching the graph of a parabola (maybe even use the graph from the warm-up Problem), and ask what important geometric features the graph has. In addition to the intercepts and the concavity (though they probably do not know this term, they might comment on the ``bowl" shape of each graph), say that an important feature of the parabola is the vertex. Point out the vertex in each example. Say the vertex is either the highest point or the lowest point on the graph of a quadratic function, depending on whether the function is concave up or concave down. Just like the standard form highlights the $y$-intercept and the factored form highlights the $x$-intercepts, we have a third way to write the equation for a quadratic function that highlights the vertex.


  • Example:






  • Example:






You may find it valuable to take this opportunity to connect the vertex form to what students learned in chapter 5. Have your groups find the explicit equation of the function whose graph is the graph of $f(x)=x^2$ translated left/right and up/down by some non-zero amount. While they do this, put up a graph of the resulting function. Have them identify the coordinates of the vertex. 

The vertex form of a quadratic function is
\[
y = a(x-h)^2 +k,
\]
where $a$ is a non-zero constant and $(h,k)$ is the vertex of the parabola. We say that the vertical line which is the graph of $x=h$ is the axis of symmetry of the parabola.


Have students do Problems 3 and 4.

Find a formula given the vertex and another point on a parabola

Discuss student solutions to Problem 4, and then produce a graph of the function in Problem 4. Use this graph to talk about how you could also take a graph and derive a formula.

Have students do Problem 5.



Complete the square

Note: If students need or want more help with completing the square, refer them to the OER.


Since completing the square is an algorithm, emphasize to students that the purpose is to convert a quadratic function in one form to vertex form. Do several examples and give thorough steps on how to complete the square.


  • Example: \vspace{1.5in}
  • Example: \vspace{1.5in}


Have students do Problems 6-7.

Quadratic functions appear in many places in the real world, highlight this again to your students. Choose at least one of problems 8, 9 and Focus problem for students to work through. Here is a cool desmos animation for problem 8, if you do it. The demo is also embedded below. If you run out of time to complete the other questions encourage students to finish the last two problems at home or in the MRC.


Comments

For the pedagogy project (or your own interest), and edited version of this lesson plan can be found here: http://www.math.unl.edu/~nwakefield2/FYM/index.php/6.2:_The_Vertex_of_a_Parabola/LailaAwadalla

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