The function 𝑦 = ax2. Its properties and its graph

Objective:

  • Know the properties and graph the quadratic function of the form 𝑦 = ax2 + 𝑏π‘₯ + 𝑐, π‘Ž β‰  0; 
  • Find values of a function from given values of an argument and find the value of an argument from given values of a function

This virtual activity is designed for use in algebra lessons on the following topics:

  • Grade 7. “functions of type 𝑦 = ax2, its graphs and properties”

Theoretical part

A function of type 𝑦 = ax2 + 𝑏π‘₯ + 𝑐, π‘Ž β‰  0 is called a quadratic function.

Properties of the quadratic function:

  • Area of definition: D(f)=R (all real numbers).
  • Set of values: E(f)=R (all real numbers).

Parity:

  • If a>0, the function is even.
  • If a<0, the function is odd.

Increasing/decreasing:

  • If a>0, the function is increasing on the interval (βˆ’ ∞;-b2a), decreasing on the interval (-b2a; ∞).
  • If a<0, the function is decreasing on the interval (βˆ’ ∞;-b2a), increasing on the interval (-b2a; ∞).

The graph of a quadratic function is a parabola.

Elements of a parabola:

  • Vertex of the parabola: point with coordinates xb=-b2a ΠΈ yb=f(b2a).
  • Axis of symmetry: a line passing through the vertex of the parabola parallel to the Oy axis.

Virtual experiment

The Graphing Quadratics simulator allows students to explore the graph of a quadratic function. On the Explore screen, students can use the sliders to explore the effect of each term of the quadratic function on the graph of the parabola. The Standard Form screen focuses on the principal point, the axis of symmetry. Students can customize the function, but values are limited to integers. 

Progression:

Unit 1: Explore

Step 1. Run the simulation: you will be presented with 4 different modes: Explore, Standard Form, Vertex Form, and Focus&Directrix. In this work, you will work with the first two modes. Run the Explore mode.

Step 2. You are given:

  • OXY coordinate plane and a graph of a parabola (1);
  • Tools to display the point values (x,y) in the coordinates of the graph (2);
  • You can keep the graph off the screen by clicking on the eye (3);
  • Equation 𝑦 = ax2 + 𝑏π‘₯ + 𝑐, the values of a, 𝑏, 𝑐, and the tools to change their values (4);
  • You can save a chart type by pressing the camera button (5);
  • You can delete a saved graph by clicking the eraser button (6);
  • If you open the quadratic terms section: the equations 𝑦 = ax2, 𝑦 = 𝑏π‘₯, 𝑦 = 𝑐 are given. You can see their graphs on the coordinate plane by checking the box for them. You can see the equations of these graphs by checking the equations button (7);
  • Reload button (8).

Step 3. a=1. Change this parameter differently. If you increase the value of a, the distance of the parabola will decrease, if you decrease it, you will see the distance increase. When you go to a negative value, the graph looks downward.

Step 4. 𝑏=0. Change this parameter differently. If you increase the value of 𝑏, the parabola moves to the left the coordinate, if you decrease it moves to the right.

Step 5. 𝑐=0. Change this parameter differently. If you increase the value of 𝑐, the parabola will move up the coordinate, if you decrease it, the parabola will move down.

Step 6. Construct different equations and study their graph by changing the values of a, 𝑏, 𝑐. You can save and explore the graph types by clicking the camera button.

Step 7. Try to explore the points of the parabola using the tools that show the point values (x,y) in the coordinates of the graph.

Step 8. Open the equations section. Include the equations 𝑦 = ax2, 𝑦 = 𝑏π‘₯, 𝑦 = 𝑐, and include the equations button. Examine the changes in the graph.

Section 2: Standard Form

Step 9. Go to Standard Form. You are given:

  • The OXY coordinate plane and a graph of a parabola;
  • Tools to display the point values (x,y) in the coordinates of the graph;
  • You can keep the graph off the screen by clicking on the eye;
  • Equation 𝑦 =  π‘¦ = ax2 + 𝑏π‘₯ + 𝑐 , the values of a, 𝑏, 𝑐, and the tools to change their values;
  • You can save the chart type by pressing the camera button;
  • You can delete a saved graph by pressing the eraser button;
  • A vertex point, an axis of symmetry, and roots are given. You can see them on the coordinate plane by checking the checkbox. You can see the equations of these graphs by checking the equations button. You can see the peak point of the graph, the axis of symmetry, and the values of the roots (x, y) using the coordinates button;
  • Reload button.

Step 10. a=1. Change this parameter. Examine the changes in the graph.  In this section, the coefficients are represented as a natural number. 

Step 11. Create different equations and explore their graphs by changing the values of 𝑏, 𝑐.

Step 12. Try to explore the points of a parabola using tools that show the point values (x, y) in the coordinates of the graph.

Step 13. Examine the coordinates of the graph by labeling the peak point, axis of symmetry, and roots.

Step 14. Try adding equation buttons. The equation belonging to the graph appears next to the graphs. 

Step 15. Do various investigations by changing the data in the equation.

Conclusion

The Graphing Quadratics simulator is a valuable tool for teaching students about graphing quadratic functions. The simulator has become interactive and visually useful by offering various tools for learning graphs such as displaying points, displaying equations, and saving graphs.