Module 2: Lesson 2

 

Self-Check 2

 

Questions 5.b. and 5.c. and questions 6.b. and 6.c. from “Practise” on page 87


    1. The two equations can be graphed as follows:

       

      This graphic shows the graphs of two functions. The first is y equals 2x plus 6, which is a linear function. The second is y equals begin the square root of 2x plus 6 end square root, which is a radical function. The graphs intersect at (negative 3, 0) and (negative 2.5, 1).

      For y = 2x + 6:

       
      Domain: {x|x ∈ R}

      Range:

      		 {y|y ∈ R}

      In graphs of square root functions, the domains of the square root functions are restricted. Any portion of the original graph that lies below the x-axis will not appear in the graph of the square root function. In this case, as seen on the graph, y < 0 occurs when x < −3. As a result, for

       
      Domain: {x|x ≥ −3, x ∈ R}

      Range:

      		 {y|y ≥ 0, y ∈ R}

    2. The two equations can be graphed as follows:

       

      This graphic shows the graphs of two functions. The first is y equals negative x plus 9. The second is y equals begin square root negative x plus 9 end square root. The graphs intersect at the points (9, 0) and (8, 1).

      For y = −x + 9:

       
      Domain: {x|x ∈ R}

      Range:

      		 {y|y ∈ R}

      In graphs of square root functions, the domains of the square root functions are restricted. Any portion of the original graph that lies below the x-axis will not appear in the square root graph. In this case, y < 0 occurs when x < 9. As a result, for

       
      Domain: {x|x ≤ 9, x ∈ R}

      Range:

      		 {y|y ≥ 0, y ∈ R}

    1. The two equations can be graphed as follows:

       

      This is a combined graph of the quadratic function y equals 2 minus x squared, and the radical function y equals begin the square root of 2 minus x squared and end the square root. The radical function is a semi-circle with its centre at (0, 0) and radius at the square root of 2. The graphs intersect at begin the negative square root of 2 end square root, followed by a comma and zero, (-1, 1), (1, 1), and begin the square root of 2, followed by a comma and a zero.

      For y = 2 − x2:

       
      Domain: {x|x ∈ R}

      Range:

      		 {y|y ≤ 2, y ∈ R}

      In graphs of square root functions, the domains of the square root functions are restricted. Any portion of the original graph that lies below the x-axis will not appear in the square root graph. In this case, the y-values are less than zero when and when This corresponds to all of the points when y < 0. The range also has an upper limit, however; in the original graph, the maximum occurs at y = 2. In the square root graph, the maximum value will be the square root of the maximum value (because this maximum value is above 0). As a result, for


       
      Domain:

      Range:


    2. The two equations can be graphed as follows:

       

      This illustration shows two graphs. One is a graph of the quadratic function y equals x squared plus 6, which is a parabola opening upward with a vertex at (0, 6) . The other graph is the radical function y equals begin square root x squared plus 6 end of the square root. There are no points of intersection.

      For y = x2 + 6:


       
      Domain: {x|x ∈ R}

      Range:

      		 {y|y ≥6, y ∈ R}

      In graphs of square root functions, the domains of the square root functions are restricted. Any portion of the original graph that lies below the x-axis will not appear in the square root graph. In this case, however, there are no points where y < 0. There is a minimum on the range in the original function, however; because this minimum is above the y-axis, the new minimum will be the square root of 6. As a result, for

       
      Domain: {x|x ∈ R}

      Range:


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