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

i. Addition

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.

Another Addition example.

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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

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

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)

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.

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