11.1: Graphing Quadratic Equations
This activity is intended to supplement Algebra I, Chapter 10, Lesson 1.
ID: 9406
Time required: 60 minutes
Topic: Quadratic Functions & Equations
- Graph \begin{align*}a\end{align*} quadratic function \begin{align*}y = ax^2 + bx + c\end{align*} and display a table for integral values of the variable.
- Graph the equation \begin{align*}y = a(x - h)^2\end{align*} for various values of \begin{align*}a\end{align*} and describe its relationship to the graph of \begin{align*}y = (x - h)^2.\end{align*}
- Determine the vertex, zeros, and the equation of the axis of symmetry of the graph \begin{align*}y = x^2 + k\end{align*} and deduce the vertex, the zeros, and the equation of the axis of symmetry of the graph of \begin{align*}y = a(x - h)^2 + k\end{align*}
Activity Overview
In this activity, students graph quadratic functions and study how the constants in the equations compare to the coordinates of the vertices and the axes of symmetry in the graphs. The first part of the activity focuses on the vertex form, while the second part focuses on the standard form. Both activities include opportunities for students to pair up and play a graphing game to test how well they really understand the equations of quadratic functions.
Teacher Preparation
This activity is designed to be used in an Algebra 1 classroom. It can also be used as review in an Algebra 2 classroom.
- Problem 1 introduces students to the vertex form of a quadratic equation, while Problem 2 introduces the standard form. You can modify the activity by working through only one of the problems.
- If you do not have a full hour to devote to the activity, work through Problem 1 on one day and then Problem 2 on the following day.
Classroom Management
- This activity is intended to be mainly teacher-led, with breaks for individual student work. Use the following pages to present the material to the class and encourage discussion. Students will follow along using their handhelds.
Associated Materials
- Student Worksheet: Graphing Quadratic Equations, http://www.ck12.org/flexr/chapter/9620
Solutions
Problem 1
- Answers will vary. Sample answer: the curve appears symmetric, and becomes less steep as \begin{align*}x\end{align*} increases or decreases.
- \begin{align*}2\end{align*}
- \begin{align*}3\end{align*}
- \begin{align*}1\end{align*}
- The lowest point of the graph is to the right of the \begin{align*}y-\end{align*}axis.
- The lowest point of the graph is to the left of the \begin{align*}y-\end{align*}axis.
- The graph moves away from the \begin{align*}y-\end{align*}axis.
- The graph moves closer to the \begin{align*}y-\end{align*}axis.
- The graph will center on the \begin{align*}y-\end{align*}axis.
- Answers will vary.
- Answers will vary.
- The \begin{align*}x-\end{align*}coordinates of the points are opposites of each other. The \begin{align*}y-\end{align*}coordinates of the points are the same.
- The left and right points are equidistant from the \begin{align*}y-\end{align*}axis.
- \begin{align*}x = 0\end{align*}
- The vertex is \begin{align*}(h, \ k).\end{align*}
- The parabola opens up.
- The parabola opens down.
- The parabola becomes “narrower.”
- The parabola becomes “wider.”
- \begin{align*}a\end{align*}
- \begin{align*}(h, \ k)\end{align*}
- \begin{align*}h\end{align*}
- Check students graphs.
- Check students graphs.
- Check students graphs.
Problem 2
- \begin{align*}-\frac{b}{2a}\end{align*}
- \begin{align*}2\end{align*}
- \begin{align*}0\end{align*}
- \begin{align*}-4\end{align*}
- \begin{align*}0\end{align*}
- \begin{align*}(0, \ 4)\end{align*}
- No
- When \begin{align*}a\end{align*} is positive, the parabola opens up. When \begin{align*}a\end{align*} is negative the parabola opens down. The greater the absolute value of \begin{align*}a\end{align*}, the “narrower” the parabola. The smaller the absolute value of \begin{align*}a\end{align*}, the “wider” the parabola.
- \begin{align*}c\end{align*} is the \begin{align*}y-\end{align*}intercept of the parabola
- Check students graphs.
- Check students graphs.
- Check students graphs.
Solutions for the Summary
1. \begin{align*}y = a(x - h)^2 + k\end{align*}
2. \begin{align*}a; (h, \ k); h\end{align*}
3. \begin{align*}y = ax^2 + bx + c\end{align*}
4. \begin{align*}-\frac{b}{2a}; \ x=-\frac{b}{2a}; \ c\end{align*}
Sketch the graph of each function. Identify the vertex and the equation of the axis of symmetry. Then check your graphs with your calculator.
5. \begin{align*}y=x^2+4\end{align*}
vertex \begin{align*}(0, \ 4)\end{align*}
axis of symmetry \begin{align*}x = 0\end{align*}
6. \begin{align*}y=(x-3)^2+5\end{align*}
vertex \begin{align*}(3, \ 5)\end{align*}
axis of symmetry \begin{align*}x = 3\end{align*}
7. \begin{align*}y=-(x-2)^2\end{align*}
vertex \begin{align*}(2, \ 0)\end{align*}
axis of symmetry \begin{align*}x = 2\end{align*}
8. \begin{align*}y=x^2+6x+9\end{align*}
vertex \begin{align*}(-3, \ 0)\end{align*}
axis of symmetry \begin{align*}x = -3\end{align*}
9. \begin{align*}y=-3x^2+6x+1\end{align*}
vertex \begin{align*}(1, \ 4)\end{align*}
axis of symmetry \begin{align*}x = 1\end{align*}
10. \begin{align*}y=x^2+1\end{align*}
vertex \begin{align*}(0, \ 1)\end{align*}
axis of symmetry \begin{align*}x = 0\end{align*}
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