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A quadratic equation is a polynomial equation of the 2nd degree. The standard form of a quadratic equation is: 0 = a x 2 + b x + c, where a, b and c are real numbers a and 0. If we replace 0 with y, we get a quadratic function y = a x 2 + b x + c, whose graph will be parabola. Sometimes, if you look at a quadratic function, you can see how it has been converted from a simple function y = x 2 . You can then plot the equation by modifying the parent plot accordingly. For example, for a positive number c, the graph y = x 2 + c is the same as the graph y = x 2 shifted up by c units. Similarly, plotting y = a x 2 stretches the plot vertically by a factor of a (negative values flip the parabola.) We can see some other transformations in the following examples. Example 1. Draw a graph of the function y = 2 x 2 - 5. If we start with y = x 2 and multiply the right side by 2, the graph will stretch vertically by 2 times. If we then subtract 5 from the right side of the equation, it moves the graph down by 5 units. Example 2: Plot the function y = - 1 2 ( x - 3 ) 2 + 2 . If we start with y = x 2 and replace x with x - 3 , this will move the graph 3 units to the right. Then when we multiply the right hand side - 1 by 2, it flips the parabola upside down and gives it a vertical contraction (or push) of 2 times. Finally, when we add 2 to the right side, it shifts the parabola. chart up 2 units. Use the sliders to change the values of the parameters in the completed square form of the equation. See how the shape of the graph changes. Plot quadratic points (1, 5) (top), (2, 6) and (3, 9) Plot quadratic points (-3, -2) (top), (-2) , 0) and (-1) , 6) First try to figure out what the equation will be, then use the applet to check the solution. Page 2 At the end of this part you will be able to: Draw quadratic functions as (f(x)=x^2+k) Draw quadratic functions as (f(x)= ( x-h)^2 )quadratic functions of the form (f(x)=ax^2) Graphic quadratic functions using transformations. Find a quadratic function on its graph Before you begin, do this readiness test. In the last section we learned how to represent quadratic functions in terms of their properties. Another method is to start with the base graph (f(x)=x^2) and move it according to what is specified in the functional equation. We call this plotting quadratic functions using transformations. In the first example we represent the quadratic function (f(x)=x^2) by points. Then we will see the effect of adding the constant (k) to the equation on the graph of the new function (f(x)=x^2+k) has. Denote (f(x)=x^2), (g(x)=x^2+2) and (h(x)=x^2-2) in this way the rectangular coordinate system itself. Describe how adding a constant to a function affects the underlying parabola. The decision. Plotting the points helps us see the effect of constants on the underlying plot (f(x)=x^2). We fill in the table for all three functions. 9.7.1. Figure The (g(x)) values are two larger than the (f(x)) values. Also, the values of (h(x)) are two smaller than the values of (f(x)). We will now represent all three functions in a single rectangular coordinate system. 9.7.2. Picture. The graph of (g(x)=x^2+2) is the same as the graph of (f(x)=x^2), but up (2) units postponed . The graph (h(x)=x^2-2) is the same as the graph (f(x)=x^2), but shifted down by (2) units. Draw (f(x)=x^2, g(x)=x^2+1,) and (h(x)=x^2-1) in the same rectangular coordinate system in vol. Describe how adding a constant to a function affects the underlying parabola. answer a Figure 9.7.3 b. The graph (g(x)=x^2+1) is the same as the graph (f(x)=x^2), but shifted up by (1) one. The graph (h(x)=x^2-1) is the same as the graph (f(x)=x^2), but shifted down by (1) one. Draw (f(x)=x^2, g(x)=x^2+6,) and (h(x)=x^2-6) in the same rectangular coordinate system in vol.How does adding a constant to a function affect the base parabola? Answer Figure 9.7.4b. The graph of (h(x)=x^2+6) is the same as the graph of (f(x)=x^2) but shifted up by (6) units. The graph (h(x)=x^2-6) is the same as the graph (f(x)=x^2) but shifted down by (6) units. The last example shows that to represent a quadratic function in the form (f(x)=x^2+k) we use the basic parabola (f(x)=x^2 ). and move it vertically ((k>0)) or down ((k<0)). This transformation is called a vertical shift. The graph (f(x)=x^2+k) moves the graph (f(x)=x^2) vertically by (k) units. If (k>0), move the parabola up by (k) units vertically. If (k<0), move the parabola vertically down by (|k|) units. Now that we have seen the effect of the constant (k), it is easy to plot functions of the form (f(x)=x^2+k). We simply start with the basic parabola (f(x)=x^2) and then move it up or down. A quick sketch exercise (f(x)=x^2) may help. We know the values and can base the graph on them. 9.7.5. image. Once you know this parabola, applying transformations is easy. The following example requires a vertical offset. Graph (f(x)=x^2-3) with vertical shift. Solution: First, we draw the graph (f(x)=x^2) on the grid. Determine (k). Move the graph (f(x)=x^2) down (3). 9.7.1. table (f(x)=x^2-5) graph with vertical shift. Answer 9.7.10. the graph of the image (f(x)=x^2+7) with a vertical shift. Answer Figure 9.7.11. In the first example, we plotted the quadratic function (f(x)=x^2) by plotting the points, and then saw the addition of the constant (k) to the function on the resulting graph. from the new function (f(x)=x^2+k). We now examine the effect of subtracting the constant (h) from (x) resulting in the new function (f(x)=(x-h)^2). Graph (f(x)=x^2, g(x)=(x-1)^2,) and (h(x)=(x+1)^2) on the same rectangle

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