In this exercise, we will be exploring parabolic graphs of the khung y = ax2 + bx + c, where a, b, và c are rational numbers. In particular, we will examine what happens to lớn the graph as we fix 2 of the values for a, b, or c, & vary the third.

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We have split it up into three parts:

Part One: Varying a

Let a vary while b & c remain fixed.

Example 1: Let b = 0 and c = 0.

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So, what"s happening? The traditional graph of y = x2 is shown in purple. We have seen that increasing the value of the nofxfans.comfficient a narrows the parabola, và changing the sign of the nofxfans.comfficient a changes the direction of the parabola.

Example 2: Let b=1 & c=0 while still varying a.

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So, what is happening now? All of these parabolas still pass through the origin, but the bases of the parabolas have moved.

Example 3: Leta vary, b = -1, and c = 0.

Based on the picture above, I would expect it to look similar lớn Example 2, except that the graph will shift to the right for the upward-shaped parabolas và to the left for the parabolas facing down. Let"s see if we"re right.

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The graph did exactly what we expected it lớn do, based on our observations in the previous graph.

Example 4: Vary a và let b = 1 và c =1.

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We see that the parabolas no longer go through the origin, but through the point (0,1) instead.

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Part 2: Varying b

Let b vary while a & c remain fixed.

Example 1: Let a = 0 và c = 0.

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I don"t think there is anything really surprising in the picture above. With both a và c equal to zero, we are left with a linear function. When b is positive, the line remains in quadrants I & III, the quadrants where both x and y have the same sign / direction. When b is negative, the line is in quadrants II and IV. Notice that these lines always go through the origin, and that when you increase b, the line gets steeper.

Example 2: Let a = 0 và c = 1.

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The graph looks exactly lượt thích the one above, except for the fact that it has shifted 1 unit in the positive direction. The graphs no longer go through the origin, but through the point (0,1). Also, notice that now the equations for the graphs are the normal linear equations of the form y = mx + b, where m is the slope andb is the y-intercept.

Example 3: Let a = 1 & c = 0.

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We now have a parabola again. Notice that the bases of the parabolas are all below the x-axis and that they still go through the origin. The graphs with positive values of b have shifted down and to the left, those with negative values of b have shifted down và to the right.

Example 4: Let a = 1 and c = 1.

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The graph looks lượt thích the one in Example 4, except for the fact that it has shifted up one unit và all parabolas go through the point (0,1).

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Part 3: Varying c

Let c vary while a and b remain fixed.

Example 1: Let a = 0 & b = 0.

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These are all horizontal lines crossing the y axis at the c value.

Example 2: Let a = 1 and b = 0.

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We have a normal parabola with the base of the parabola on the y-axis at the value of c.

Example 3: Let a = 0 and b = 1.

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All of these graphs have a slope of 1 (the b value) and cross the y-axis at the c value.

Example 4: Let a = 1 và b = 1.

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We have a series of parabolas whose bases are lớn the left of the y-axis and that cross the y-axis at the c value.