Quadratic function and its graph

Objective:

  • To know the properties and graphs of quadratic functions of the form 𝑦 =a(x-m)2 , 𝑦 =ax2+n, 𝑦 =a(x+m)2+n, π‘Ž β‰  0;
  • 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 8. “Quadratic function and its graph”

Theoretical part

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

The graph of the function y=f(x)+m is obtained by moving the graph of the functiony=f(x) vertically:

  • upward if m is a positive number;
  • down if m is a negative number.

The graph of the functiony=f(x+n) can be obtained by moving the graph of the function y=f(x) horizontally:

  • to the left by n units if n is a positive number;
  • to the right by n units if n is a negative number.

The graph of the function y=f(x+n)+m can be obtained from the graph of the function y=f(x) by performing two consecutive moves:

  1. Horizontal move:
    • To the left by n units if n is a positive number;
    • to the right by n units if n is a negative number.
  2. Vertical move:
    • Up by m units if m is a positive number;
    • down by m units if m is a negative number.

The standard form of a quadratic equation is as follows: ax2 + 𝑏π‘₯ + 𝑐.

The shape of the vertex of the quadratic equation is (a(x-h)2+k, where (a) is a constant that determines whether the parabola opens up or down, and ((h,k)) – are the coordinates of the vertex of the parabola. For example:

  • (2(x-7)2+3); the vertex is at (7, 3).
  • (2(x+7)2-3); the vertex is at the point (-7, -3).

The graph of a quadratic function is a parabola.

Elements of a parabola:

  • Vertex of the parabola: point with coordinatesxb=-b2a ΠΈ yb=f(b2a).
  • Axis of symmetry: a line passing through the vertex of the parabola parallel to the Oy axis.
  • Directrix: a line symmetric to the axis of symmetry with respect to the vertex of the parabola.
  • Focus: a point whose distance from any point of the parabola is equal to the distance from that point to the directrix.

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. On the Vertex Form screen, students explore the transformation of a parabola and identify the relationship between the graph of a parabola and a quadratic function. In the Focus&Directrix screen, students create a parabola based on the peak and focus.

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. 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. Construct and study the graph of various equations by changing the values of a, 𝑏, 𝑐. In this section, the coefficients are represented as a natural number.

Step 11. Try to study the points of the parabola using the tools that show the point values (x, y) in the coordinates of the graph.

Step 12.Try adding the peak point, axis of symmetry, roots, and equations buttons. Examine the changes in the graph.

Section 3: Vertex Shape

Step 13. Go to Vertex Form. You are given:

  • 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 = (a(x-h)2+k, the values of a, h,k and the tools to change their value;
  • You can save the graph type by pressing the camera button;
  • You can delete a saved graph by pressing the eraser button;
  • A vertex point, axis of symmetry 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 values of the vertex point of the graph and the axis of symmetry (x, y) using the coordinates button;
  • Reload button.

Step 14. Construct and examine the graphs of various equations by changing the values of a, h,k. In this section, the coefficients are represented as a natural number.

Step 15. Try to explore the points of a parabola using tools that show the point values (x, y) in the coordinates of the graph.

Step 16. Try adding the peak point, axis of symmetry, and equations buttons. Examine the changes in the graph.

Section 4: Focus and Directrix

Step 17. Go to Focus&Directrix. 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;
  • The equation y=14p(x-h)2+k,, the values of p, h,k and the tools to change their value;
  • You can save the graph type by clicking the camera button;
  • You can delete a saved graph by pressing the eraser button;
  • The peak point, focus, directrix, and point of a parabola 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 values of the vertex point of the graph and the axis of symmetry (x, y) using the coordinates button;
  • Reload button.

Step 18. Construct and examine the graphs of the various equations by changing the values of p, h,k. In this section, the coefficients are represented as a natural number.

Step 19. Try to study the points of a parabola using tools that show the point values (x, y) in the coordinates of the graph.

Step 20. Try adding the peak point, focus, directrix, parabola, and equations buttons. Examine the changes in the graph.

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.