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- https://math.libretexts.org/Courses/Northeast_Wisconsin_Technical_College/College_Algebra_(NWTC)/03%3A_Polynomial_Functions/3.06%3A_Complex_Zeros/3.6E%3A_Complex_Zeros_(Exercises)29. \(f(x) = x^3 + 6x^2 + 6x + 5 = (x + 5) (x^2 + x + 1) = (x + 5)(x - (-\dfrac{1}{2} + \dfrac{\sqrt{3}}{2}i))(x - (-\dfrac{1}{2} - \dfrac{\sqrt{3}}{2}i))\) Zeros: \(x = -5\), \(x = -\dfrac{1}{2} \pm ...29. \(f(x) = x^3 + 6x^2 + 6x + 5 = (x + 5) (x^2 + x + 1) = (x + 5)(x - (-\dfrac{1}{2} + \dfrac{\sqrt{3}}{2}i))(x - (-\dfrac{1}{2} - \dfrac{\sqrt{3}}{2}i))\) Zeros: \(x = -5\), \(x = -\dfrac{1}{2} \pm \dfrac{\sqrt{3}}{2} i\) 37. \(f(x) = x^4 + x^3 + 7x^2 + 9x - 18 = (x + 2) (x - 1)(x^2 + 9) = (x + 2)(x - 1)(x + 3i)(x - 3i)\) Zeros: \(x = -2, 1, \pm 3i\) 41. \(f(x) = x^4 + 9x^2 + 20 = (x^2 + 4)(x^2 + 5) = (x - 2i)(x + 2i)(x - i\sqrt{5}) (x + i\sqrt{5})\) Zeros: \(x = \pm 2i, \pm i \sqrt{5}\)
- https://math.libretexts.org/Courses/Northeast_Wisconsin_Technical_College/College_Algebra_(NWTC)/02%3A_Linear_Functions/2.02%3A_Graphs_of_Linear_FunctionsRemember the initial value of the function is the output when the input is zero, so in the equation \(f(x)=b+mx\), the graph includes the point \((0, b)\). From a graph of a line, this tells us that i...Remember the initial value of the function is the output when the input is zero, so in the equation \(f(x)=b+mx\), the graph includes the point \((0, b)\). From a graph of a line, this tells us that if we divide the vertical difference, or rise, of the function outputs by the horizontal difference, or run, of the inputs, we will obtain the rate of change, also called slope of the line.
- https://math.libretexts.org/Courses/Northeast_Wisconsin_Technical_College/College_Algebra_(NWTC)/01%3A_Functions/1.03%3A_Rates_of_Change_and_Behavior_of_GraphsSince functions represent how an output quantity varies with an input quantity, it is natural to ask about the rate at which the values of the function are changing.
- https://math.libretexts.org/Courses/Northeast_Wisconsin_Technical_College/College_Algebra_(NWTC)/04%3A_Rational_and_Radical_Functions/4.01%3A_Rational_FunctionsIn this section, we explore functions based on power functions with negative integer powers, called rational functions.
- https://math.libretexts.org/Courses/Northeast_Wisconsin_Technical_College/College_Algebra_(NWTC)/02%3A_Linear_Functions/2.01%3A_Linear_FunctionsGiven two values for the input,\(x_{1} {\rm \; and\; }x_{2}\), and two corresponding values for the output, \(y_{1} {\rm \; and\; }y_{2}\), or a set of points, \((x_{1} {\rm ,\; \; }y_{1} )\) and\((x_...Given two values for the input,\(x_{1} {\rm \; and\; }x_{2}\), and two corresponding values for the output, \(y_{1} {\rm \; and\; }y_{2}\), or a set of points, \((x_{1} {\rm ,\; \; }y_{1} )\) and\((x_{2} {\rm ,\; \; }y_{2} )\), if we wish to find a linear function that contains both points we can calculate the rate of change, m:
- https://math.libretexts.org/Courses/Northeast_Wisconsin_Technical_College/College_Algebra_(NWTC)/01%3A_Functions/1.05%3A_Transformation_of_FunctionsThere are systematic ways to shift, stretch, compress, flip and combine functions to help them become better models for the problems we are trying to solve. We can transform what we already know into ...There are systematic ways to shift, stretch, compress, flip and combine functions to help them become better models for the problems we are trying to solve. We can transform what we already know into what we need, hence the name, “Transformation of functions.” When we have a story problem, formula, graph, or table, we can then transform that function in a variety of ways to form new functions.
- https://math.libretexts.org/Courses/Northeast_Wisconsin_Technical_College/College_Algebra_(NWTC)/01%3A_Functions/1.02%3A_Domain_and_RangeOne of our main goals in mathematics is to model the real world with mathematical functions. In doing so, it is important to keep in mind the limitations of those models we create.
- https://math.libretexts.org/Bookshelves/Precalculus/Book%3A_Precalculus__An_Investigation_of_Functions_(Lippman_and_Rasmussen)/05%3A_Trigonometric_Functions_of_Angles/5.04%3A_The_Other_Trigonometric_FunctionsIn the previous section, we defined the sine and cosine functions as ratios of the sides of a right triangle in a circle. Since the triangle has 3 sides there are 6 possible combinations of ratios. Wh...In the previous section, we defined the sine and cosine functions as ratios of the sides of a right triangle in a circle. Since the triangle has 3 sides there are 6 possible combinations of ratios. While the sine and cosine are the two prominent ratios that can be formed, there are four others, and together they define the 6 trigonometric functions.
- https://math.libretexts.org/Bookshelves/Precalculus/Book%3A_Precalculus__An_Investigation_of_Functions_(Lippman_and_Rasmussen)/06%3A_Periodic_Functions/6.01%3A_Sinusoidal_GraphsIn this section, we will work to sketch a graph of a rider’s height above the ground over time and express this height as a function of time.
- https://math.libretexts.org/Bookshelves/Precalculus/Book%3A_Precalculus__An_Investigation_of_Functions_(Lippman_and_Rasmussen)/06%3A_Periodic_Functions/6.01%3A_Sinusoidal_Graphs/6.1E%3A_Sinusoidal_Graphs_(Exercises)For the graphs below, determine the amplitude, midline, and period, then find a formula for the function. Suppose you know the temperature is 50 degrees at midnight and the high and low temperature du...For the graphs below, determine the amplitude, midline, and period, then find a formula for the function. Suppose you know the temperature is 50 degrees at midnight and the high and low temperature during the day are 57 and 43 degrees, respectively. Suppose you know the temperature is 68 degrees at midnight and the high and low temperature during the day are 80 and 56 degrees, respectively. Assuming t is the number of hours since midnight, find a function for the temperature, D, in terms of t.
- https://math.libretexts.org/Bookshelves/Precalculus/Book%3A_Precalculus__An_Investigation_of_Functions_(Lippman_and_Rasmussen)/03%3A_Polynomial_and_Rational_Functions./3.01%3A_Power_Functions/3.1E%3A_Power_Functions_(Exercises)Find the long run behavior of each function as \(x \to \infty\) and \(x \to -\infty\) As \(x \to \infty\), \(f(x) \to \infty\) As \(x \to -\infty\), \(f(x) \to -\infty\) As \(x \to \infty\), \(f(x) \t...Find the long run behavior of each function as \(x \to \infty\) and \(x \to -\infty\) As \(x \to \infty\), \(f(x) \to \infty\) As \(x \to -\infty\), \(f(x) \to -\infty\) As \(x \to \infty\), \(f(x) \to -\infty\) As \(x \to -\infty\), \(f(x) \to \infty\) As \(x \to \infty\), \(f(x) \to -\infty\) As \(x \to -\infty\), \(f(x) \to -\infty\) As \(x \to \infty\), \(f(x) \to \infty\) As \(x \to -\infty\), \(f(x) \to \infty\)
