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Chapter 4

# Chapter 4 - Chapter4:LinearFunctions(4.1...

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Chapter 4: Linear Functions (4.1) Definition: A linear function (always degree 1) is defined by an equation of the form f ( x ) = Ax + B , where A and B are constants is called the Standard form. For a Line passing through (x 1 ,y 1 ) and (x 2 ,y 2): Slope= 1) Slope‐intercept form of Line: 2) Point‐slope Form: 3) Standard Form: Ex: If f(3) = 2 and f(‐3) = ‐4, find the linear function defining f(x) in the slope‐ intercept form.

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Application of linear function concept: A factory owner buys a new machine for \$20, 000. After eight years, the machine has a salvage value of \$1000. Find a formula for the value of the machine after t years when 0 t 8 . Application of linear function in economics: Suppose the cost (\$) to a manufacturer of producing x units of a certain motorcycle is given by Cost function C(x) = 220x + 4000 (y = mx +b form) where m = 220 and b = 4000 a) Find the marginal cost = additional cost to produce one more unit b) Compute the cost of 500 motorcycles c) Use the results from a) and b) to find the cost of 501 motorcycles
d) Find the fixed cost (the cost before starting production) Ex: Let x denote a temperature on the Celsius scale, and let y denote the corresponding temperature on the Fahrenheit scale. a) Find the linear function relating x and y; use the facts that 32 0 F corresponds to 0 0 C and 212 0 F corresponds to 100 0 C. Write the function in standard form. b) What Celsius temperature corresponds to 98.6 0 F? c) Find a number z for which z 0 F=z 0 C.

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Quadratic Function (4.2) Definition: A quadratic function (always degree 2) is defined by an equation of the form f ( x ) = ax 2 + bx + c where a, b, c are constants, with a 0 . The graph of a quadratic functions is a parabola. Graphs representing a quadratic equation: Terms Vertex: Axis of symmetry Vertex form of a parabola: f ( x ) = y = a ( x h ) 2 + k Ex: Rewrite the function f ( x ) = x 2 + 6 x 3 in the vertex form. Completing the square method:
Using the vertex formula: Note: The maximum or minimum value of f(x) occurs at the vertex. In general given a quadratic function of the form y = f ( x ) = ax 2 + bx + c , express it in the vertex form by completing the square.

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How does the constant a plays a role? Now let’s go back to the first example to find the x‐intercept, y‐intercept and the general shape of the parabola. Graph the parabola and note how the value and sign of the constant a determine the shape.
Ex: Graph the parabola f ( x ) = 2 x 2 + 20 x + 15 by determining its vertex, y‐intercept, axis of symmetry. Does the graph open up or down and why? Find the vertex by completing the square method: Check your answer from above by using the vertex formula:

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x‐intercept: y‐intercept: Graph:
Setting up Equations that define functions (4.4) Steps: 1) After reading the problem carefully draw a picture or make a table that conveys the given information. Figure out

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Chapter 4 - Chapter4:LinearFunctions(4.1...

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