### Mike Cotton

In this assignment, a pair of linear functions f(x) & g(x) will be investigated under addition, multiplication, division, and composition.

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In this example of addition, the constant value in the first linear function is varied from 3 to -3. Note that the sum of the individual slopes is the resulting slope, and that the sum of the y-intercepts is the resulting y-intercept [i.e. (ax + b) + (cx + d)=(a + c)x + (b + d)]. The result of the addition of the above functions is that the resulting slope remains constant, and the y-intercept changes. The same result would occur if the constant value in the second linear function is varied.

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In this example of addition, the slope value in the first linear function is varied from 4 to -2. Again, note that the sum of the individual slopes is the resulting slope, and that the sum of the y-intercepts is the resulting y-intercept [i.e. (ax + b) + (cx + d)=(a + c)x + (b + d)]. The result of the addition of the above functions is that the resulting slope changes, and the y-intercept remains constant. The same result would occur if the slope value in the second linear function is varied. Note that if the slope and constant values are both varied, both the slope and the y-intercept will vary.

ii. Multiplication

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In this example of multiplication, the constant value of the first linear function was varied from -3 to 1. The result of multiplying two linear functions together is a quadratic equation. A quadratic equation is a second-order polynomial equation in a single variable x, which is generally of the form ax2+bx+c = 0. The second representation of the equations above (after the multiplication) is of the form 4p(y – k) = (x - h)2, where h is the x-coordinate of the vertex of the parabola, and k is the y-coordinate of the vertex. Therefore the point V(h, k) is the vertex of each parabola. As you can see above this point is different for each equation. Note that all of the parabolas above are similar. This is because 4p = -0.125 (p = -0.03125) in all of the equations. If p were positive the parabolas would open upwards.

Using the first equation above, it is worth pointing out that the x-intercepts (roots) of the quadratic are the same as the x-intercepts of the two original linear functions.

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Another multiplication example

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In this example of multiplication, the slope value in the second linear function was varied from -3 to -1. Again, the result of multiplying two linear functions together is a quadratic equation. The second representation of the equations above (after the multiplication) is of the form 4p(y – k) = (x - h)2. Note that all of the parabolas above are not similar. This is because 4p a different value in all of the equations above. If p were positive the parabolas would open upwards. Again, the x-intercepts (roots) of the quadratic are the same as the x-intercepts of the two original linear functions
iii. Division

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In this example of division of two linear functions, there are three different results. The result of the division of the first two equations (purple & red) is a constant. This is because the numerator is a (constant) multiple of the denomerator. The next two equations (blue & green) result in hyperbolas (described in more detail below). The last equation (lite blue) results in a line with a slope that is not equal to zero (i.e. y = ax + b).

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As mentioned above, one result of the division of the two linear functions is a hyperbola. Notice that the vertical asymptote is the value for x when the denomerator is set equal to zero, and that the x-intercept of the hyperbola is the value for x when the numerator is se equal to zero.

iv. Composition

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The composition of two linear functions is expected to be of the form: a(cx + d) + b = acx + (ad + b). This result is another linear function.