Function and graph of a function

Objective:

  • To grasp the concepts of function and graph of a function; 
  • Know the definition of the function 𝑦 = 𝑘𝑥, construct its graph; 
  • Know the definition of the linear function 𝑦 = 𝑘𝑥 + 𝑏, graph it;
  • Construct the graph of the function y=ax2 (a≠0); 
  • Construct the graph of the function y=ax3 (a≠0).

This virtual activity is designed to be used in the algebra lesson in the next chapter:

  • Grade 7. Linear function and its graph

Theoretical part

The function 𝑦 = 𝑘𝑥

The linear function 𝑦 = 𝑘𝑥 + 𝑏 is a function whose graph is a straight line.

The coefficient k determines the slope of the line.

  • If k > 0, the line is sloped upward to the right.
  • If k < 0, the line is sloped downward to the left.
  • If k = 0, the line is horizontal.

The free term 𝑏 determines the position of the line on the y-axis.

  • If 𝑏 > 0, the line intersects the y-axis at a point with a positive ordinate.
  • If 𝑏 < 0, the line intersects the y-axis at a point with a negative ordinate.
  • If 𝑏 = 0, the line passes through the origin.

Function y=ax2

The quadratic function y=ax2 is a function whose graph is a parabola.

The coefficient a determines the shape of the parabola.

  • If a > 0, the parabola is open upward.
  • If a < 0, the parabola is open downward.

The vertex of the parabola is at the point x=-b/2a.

Function y=ax3

The cubic function y=ax3 is a function whose graph is a cubic parabola.

The coefficient a determines the shape of the cubic parabola.

  • If a > 0, then the cubic parabola is open upward.
  • If a < 0, the cubic parabola is open downward.

Virtual experiment

The virtual simulation is designed to familiarize and work with the functions 𝑦 = 𝑘𝑥, y=ax2, y=ax3. This allows students to easily and clearly create graphs of these functions, as well as study their properties.

Workflow:

Part 1. Function y=kx

Step 1. Run the simulation. You are given: 

  • OXY coordinate plane (1);
  • Deviation board (2);
  • A panel of buttons: curve, residuals, values (3);
  • A set of points for plotting the graph (4);
  • Reload button (5).

Step 2. Check the function curve. You will have additional button panel: linear, quadratic, cubic equation buttons, best fit, adjustable fit. 

Step 3. Since you don’t need the deviation panel in your work, you can click the “-” button and assemble it. And the use of the residuals and values buttons is not unnecessary.

Step 4. Plot the 2 points on the coordinate plane. You will get the graph of the linear function y=kx+b. You can see the graphical equation at the top of the plane.

Step 5. You can set the values of the points (x, y) by checking the checkbox on the coordinate button. 

Step 6. Examine the change in the graph and its equation by moving the points to different locations on the plane. 

Part 2. Function 𝑦 = ax2

Step 7. Change the type of equation linear to quadratic.

Step 8. Place the third point in the plane. You will have a parabola corresponding to the function 𝑦 = ax2 + 𝑏𝑥 + 𝑐. Study the equation of the parabola.

Step 9. Examine the change in the graph and its equation by moving the points to different locations in the plane. 

Part 3. Function y=ax3

Step 10. Change the type of equation from quadratic to cubic.

Step 11. Plot the fourth point in the plane. You will have a cubic parabola corresponding to the function y=ax3+bx2+cx+d. Study the equation of the cubic parabola.

Step 12. Study the change in the graph and its equation by moving the points to different locations in the plane.

Conclusion

The virtual work described the process of graphing the linear function y=kx+b in the first section, the quadratic function 𝑦 = ax2 + 𝑏𝑥 + 𝑐 in the second section, and the cubic function y=ax3+bx2+cx+d in the third section. This simulation allowed students to extend their knowledge of the topic of function and the graph of a function by visualizing the effects of the location of points on the graph and equation in the plane.