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Linear Function
***(keep in mind cards #17-29 will not be covered)
A function of the form f(x) = Ax + B, where A and B are constants
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Slope formula
y2-y1/x2-x1
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Formula to find the linear function f(x) = Ax + B
y-y1 = m(x-x1), where m is the slope
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In the context of linear functions, we often suppress the word "average" and refer simply to the rate of change. Explain
No matter what two points we choose on the graph of a linear function, the slope will always be the same.
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cost function
a function that gives the cost C(x) for producing x units of a commodity
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marginal cost
Additional cost of producing one more unit
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What is the marginal cost in a linear function
marginal cost = the slope
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After the linear functions, the next simplest functions are the quadratic functions, which are defined by equations of the form:
f(x) = ax2 + bx + c (a ≠ 0), where a, b and c are constants and a is not zero
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The graph of any quadratic function is a curve called a _______ that is similar in shape to the basic y = __ graph.
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For quadratic functions, the parabola opens _______ when a > 0 and _______ when a < 0. The turning point on the parabola is called the ______. The axis of symmetry of the parabola y = ax2 + bx + c is the vertical line that passes through the _______
- upward
- downward
- vertex
- vertex
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Graph the function y = x2 - 2x + 3, by completing the square.
- Step 1:
- y = (x2 - 2x )+ 3
- y = (x2 - 2x + 1)+ 3 - 1 *(2x/2x)2 = 1
- y = (x - 1)2 +2
- Step 2: find y intercept by inserting 0 as your x value which gives you (0,3)
- Step 3: use the axis of symmetry from the vertex to mirror the y intercept to give you (2,3) which gives you a reasonable graph
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The parabola y = ax2 is narrower than y = x2 when? and wider when?
- narrower if |a| > 1
- wider if |a| < 1
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Parallel lines have the same _____
slope
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What can be said of the slope of perpendicular lines?
they are negative reciprocals
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First difference
Second difference
- First difference: a new list formed by subtracting adjacent members of a given list
- Second difference: a 2nd list formed by subtracting adjacent members of the first difference list
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Data is generated by a linear function if and only if the ______ differences of the ___ values are constant. It is generated by a quadratic function if and only if the _____ differences of the ____ values are constant
- first differences
- y values
- second differences
- y values
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Fixed Point
A fixed point of a function f is an input x in the domain of f such that f(x) = x (for example f(x) = 3x - 2
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Define the fixed point geometrically and state what is inferred when there is no solution
it is the x-coordinate of a point where the graph of the given function intersects the line y = x, if there is no solution, it means the graph of that function doesn't intersect the line y = x
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If (a) is a fixed point, then all of the subsequent _______ of (a) will also equal a. State an example using f(a)
- iterates
- f(a) = a ∴ f(f(a)) = a...
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In the case of f(x) = 1/2x + 2 with x0 = 1, we say the iterates of x0 = 1 _______ the fixed point 4 (4,4) and that htis target value 4 is an ______ ______ _______ of the function f
- approach
- attracting fixed point
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Repelling fixed point
the iterates of an x0 move further away from the line y = x
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The equation f(x) = kx(1 - x), is generally used to study ______ ______.
population growth
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In the equation f(x) = kx(1 - x), we assume that the population size is measured by a number between ___ and ___, where ___ corresponds to the maximum possible population size in the given environment and ___ corresponds to the case in which the population has become extinct.
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We start with a given input x0 (________) that represents the fraction of the maximum population size that is initially present. For instance, if the maximum catfish population in a pond is 100 and initially there were 70 catfish, then we would have x0 = _____
**first question is an inequality representation
- x0 (0 ≤ x0 ≤ 1)
- x0 = 70/100 = 0.7
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The next assumption using f(x) = kx(1 - x) to model population size is that the iterates of x0 represent the fraction of the maximum possible population present after each successive time interval. Explain
- f(x0) = x1 is the fraction of the maximum population after the first time interval
- f(x1) = x2 is the fraction of the maximum population after the second time interval and in general:
- f(xn-1) = xn is the fraction of the maximum population after the nth time interval
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It is important to note that the function f(x) does not represent the size of the population. The population size after n time intervals is given by:
xn * (the maximum population) = f(xn-1) * (maximum population)
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In the equation f(x) = kx(1 - x), what is k?
k is a growth parameter that is a constant; it is related to the rate of growth of the particular population. It can be for example, the fecundity of the fish in a pond, the propensity of the population not just to boom but also to bust
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How do find x0?
x0 = initial population/maximum population
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How do find the equilibrium population?
The fixed point of interest multiplied by the maximum population
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Steps for setting up equations that define functions (4)
- After reading the problem carefully, draw a picture that conveys the given information
- State in own words, as specifically as you can, what the problem is asking for (usually requires rereading the problem). Assuming that the problem asks you to find a particular quantity (or a formula for a particular quantity), assign a variable to denote that key quantity
- Label any other quantities in your figure that appear relevant. Are there equations relating these quantities?
- Find an equation involving the key variable that you identified in step 2. Now, substitute in this equation using the auxiliary equations from step 3 to obtain an equation involving only the required variables
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Distance formula
√(x2-x1)2+(y2-y1)2
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Area of triangle
A = 1/2bh
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Area of an equilateral triangle
- A = (1/2x)(x√3/2)
- A = (x2√3)/4
- *where x is the length of one of the equilateral legs
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4.5] Two numbers add to 9. What is the largest possible value for their product? How do you go about solving this?
- Call the two numbers (x) & (9-x)
- Get their products: 9x - x2
- Calculate the vertex by completing the square: (4.5, 20.25)
- Depending on if the parabola faces upward or downward, you will be looking for either a maximum or a minimum.
- In this case, it faces downward and thus gives a maximum
- Lastly, your y value (20.25) is the largest possible product
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Finding the maximum and minimum of a graph involving quadratic equations will likely involve _______ the _____. Name another situation where this method can be used.
- completing the square
- cube roots of a quadratic equation
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Explain why completing the square can be used to determine the maximum and minimum of the cube root of a quadratic equation
Although the shape may change slightly, the x value of the vertex and direction/position of the graph remains the same. The Y value needs to be cube rooted for it to correspond properly
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How do you calculate the x value of a vertex in a quadratic equation without completing the square or approximating with a graphing utility?
Use the vertex formula: x = -b/2a
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Show the proof for the vertex formula
y = ax2 + bx + c
- The parabola y = ax2 + bx + c will have the same x-coordinate for its vertex as y = ax2 + bx because the two graphs are just vertical translates of each other ∴
- ax2 + bx = 0
- x(ax + b) = 0
- x = 0 | ax + b = 0
- x = 0 | x = -b/a
- The x value that is half between 0 and -b/a will represent the proper vertex. This will be 1/2(-b/a) = -b/2a
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If asked to find the input that minimizes y = √f(x) find the input that minimizes _______. The outputs of the two functions are _______; but the same ______ serves to minimize both functions. This is provided that _____ is non-negative for all x
- the quadratic function y = f(x)
- different
- input
- f(x)
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When asked to find the point on the curve y = √x closest to the point (1,0), what 3 formulas do you rely on
- Distance formula = √(x2-x1)2+(y2-y1)2 y = √x
- x = -b/2a
- *when it just asks for the distance all you should need is the 1st two
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Revenue formula
R = number of units(x) * price per unit
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When asked for the area of a square inscribed within a square, and you are given dimensions for one of the triangles formed subsequently, how do you find the area of the smaller square?
Use area of the triangle * 4 for four triangles formed (4*1/2bh). The area of those four triangles will be equivalent to the area of the smaller square.
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