Mathematics Content Expectations
Form A: Math Alignment Table | |||||
Alignment to Math High School Content Expectations | |||||
Math High School Content Expectations | Prealgebra Math 050 to Summer 2006 |
Prealgebra Math 050 to Fall 2006 |
Introductory Algebra Math 107 Summer and Fall 2006 |
Math 112 | ACCUPLACER Tests |
A2.3 Families of Functions (linear, quadratic, polynomial, rational, power, exponential, logarithmic, and trigonometric) |
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A2.3.1 Identify a function as a member of a
family of functions based on its symbolic, or graphical representation; recognize that different families of functions have different asymptotic behavior at infinity, and describe these behaviors. |
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A2.3.2 Describe the tabular pattern associated
with functions having constant rate of change (linear); or variable rates of change. |
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A2.3.3 Write the general symbolic forms that characterize each family of functions. e.g., f (x ) = A0ax ; f (x ) = AsinBx. |
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A2.4 Lines and Linear Functions | |||||
A2.4.1 Write the symbolic forms of linear
functions (standard [i.e., Ax + By = C, where B ≠ 0 ], pointslope, and slope-intercept) given appropriate information, and convert between forms. |
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A2.4.2 Graph lines (including those of the form x
= h and y = k ) given appropriate information. |
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A2.4.3 Relate the coefficients in a linear
function to the slope and x - and y -intercepts of its graph. |
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A2.4.4 Find an equation of the line parallel or perpendicular to given line, through a given point; understand and use the facts that non-vertical parallel lines have equal slopes, and that non-vertical perpendicular lines have slopes that multiply to give - 1. |
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A2.5 Exponential and Logarithmic Functions | |||||
A2.5.1 Write the symbolic form and sketch the
graph of an exponential function given appropriate information. (e.g., given an initial value of 4 and a rate of growth of 1.5, write f (x ) = 4 (1.5)x ). |
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A2.5.2 Interpret the symbolic forms and recognize the graphs of exponential and logarithmic functions (e.g., f (x ) = 10 x , f (x ) = log x , f (x ) = ex , f (x ) = ln x ); recognize the logarithmic function as the inverse of the exponential function. |
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A2.5.3 Apply properties of exponential and logarithmic functions (e.g., ax+y = axa y ; log(ab )= log a + log b ). |
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A2.5.4 Understand and use the fact that the base
of an exponential function determines whether the function increases or decreases and understand how the base affects the rate of growth or decay. |
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A2.5.5 Relate exponential and logarithmic
functions to real phenomena , including half-life and doubling time. |
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A2.6 Quadratic Functions | |||||
A2.6.1 Write the symbolic form and sketch the
graph of a quadratic function given appropriate information (e.g., vertex, intercepts, etc.). |
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A2.6.2 Identify the elements of a parabola
(vertex, axis of symmetry, direction of opening) given its symbolic form or its graph, and relate these elements to the coefficient(s) of the symbolic form of the function. |
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A2.6.3 Convert quadratic functions from standard
to vertex form by completing the square. |
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A2.6.4 Relate the number of real solutions of a quadratic equation to the graph of the associated quadratic function. |
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A2.6.5 Express quadratic functions in vertex form
to identify their maxima or minima, and in factored form to identify their zeros. |
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A2.6.5 Express quadratic functions in vertex form
to identify their maxima or minima, and in factored form to identify their zeros. |
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A2.7 Power Functions (including roots, cubics, quartics, etc.) |
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A2.7.1 Write the symbolic form and sketch the
graph of power functions. |
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A2.7.2 Express direct and inverse relationships
as functions (e.g., y = kxn and y = kx-n , n > 0) and recognize their characteristics (e.g., in y = x^{3} , note that doubling x results in multiplying y by a factor of 8). |
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A2.7.3 Analyze the graphs of power functions, noting reflectional or rotational symmetry. |
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A2.8 Polynomial Functions | |||||
A2.8.1 Write the symbolic form and sketch the
graph of simple polynomial functions. |
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A2.8.2 Understand the effects of degree, leading coefficient, and number of real zeros on the graphs of polynomial functions of degree greater than 2. |
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A2.8.3 Determine the maximum possible number of zeros of a polynomial function, and understand the relationship between the x -intercepts of the graph and the factored form of the function. |
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A2.9 Rational Functions | |||||
A2.9.1 Write the symbolic form and sketch the
graph of simple rational functions. |
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A2.9.2 Analyze graphs of simple rational
functions (e.g., ) and understand the relationship between the zeros of the numerator and denominator and the function’s intercepts, asymptotes, and domain. |
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A2.10 Trigonometric Functions | |||||
A2.10.1 Use the unit circle to define sine and
cosine; approximate values of sine and cosine (e.g., sin 3 , or cos 0.5 ); use sine and cosine to define the remaining trigonometric functions; explain why the trigonometric functions are periodic. |
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A2.10.2 Use the relationship between degree and radian measures to solve problems. |
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A2.10.3 Use the unit circle to determine the
exact values of sine and cosine, for integer multiples of ⁄6 and ⁄4. |
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A2.10.4 Graph the sine and cosine functions; analyze graphs by noting domain, range, period, amplitude, and location of maxima and minima. |
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A2.10.5 Graph transformations of basic trigonometric functions (involving changes in period, amplitude, and midline) and understand the relationship between constants in the formula and the transformed graph. |
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STANDARD A3: MATHEMATICAL MODELING Students construct or select a function to model a real-world situation in order to solve applied problems. They draw on their knowledge of families of functions to do so. |
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A3.1 Models of Real-world Situations Using Families of Functions. Example: An initial population of 300 people grows at 2% per year. What will the population be in 10 years? |
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A3.1.1 Identify the family of function best
suited for modeling a given real-world situation (e.g., quadratic functions for motion of an object under the force of gravity; exponential functions for compound interest; trigonometric functions for periodic phenomena. In the example above, recognize that the appropriate general function is exponential (P = P_{0}a^{t} ) |
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A3.1.2 Adapt the general symbolic form of a
function to one that fits the specifications of a given situation by using the information to replace arbitrary constants with numbers. In the example above, substitute the given values P_{0} = 300 and a = 1.02 to obtain P = 300(1.02)^{t} . |
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A3.1.3 Using the adapted general symbolic form, draw reasonable conclusions about the situation being modeled. In the example above, the exact solution is 365.698, but for this problem an appropriate approximation is 365. |
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