Examples for

Common Core Math: High School Functions: Interpreting Functions

In high school, students study linear, quadratic, polynomial, exponential, logarithmic, rational and trigonometric functions. These functions can be represented with equations, words, graphs, diagrams, sequences or tables. Each time students discover a new type of function, students learn to perform relevant computations for locating key features such as x-intercepts (also called "roots" or "zeros"), y-intercepts, maxima and minima, end behavior, discontinuities, monotonicity, periodicity and more.
Common Core Standards

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CCSS.Math.Content.HSF-IF.A.1
C.7a common core interpreting functions standard

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interpreting functions common core standards
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Concept of a Function

Determine key features of functions represented by an equation, graph or table.

Determine the domain and range of a function (CCSS.Math.Content.HSF-IF.A.1):

domain and range of y = x^2 + 4x + 3
range of f(x) = 2^x - 1
domain of y = sqrt(x)
g(x) = (x^2 - x - 20)/(x^2 + 12x + 32) domain and range
log10(x) domain and range

Graph a function given its equation (CCSS.Math.Content.HSF-IF.A.1):

graph f(x) = -0.5x + 0.75
plot g(x) = (x + 5) (x - 2) (x + 2)^2
graph of y = 4 sin(2x - pi) + 3

Evaluate functions (CCSS.Math.Content.HSF-IF.A.2):

f(4) if f(x) = x^2 + 2x - 9
compute -x^2 + 2(x + 7) - 19 where x = 7
g(2) for g(x) = 3^(-x) - 10
(x - 1) * (x + 3)^2 + 7 for x = 1/2
evaluate (x + 2)/((x - 2) (x + 3)) at x = 1

Express sequences as functions (CCSS.Math.Content.HSF-IF.A.2):

1, 8, 15, 22, 29, ...
3, 5, 9, 17, 33, ...
Fibonacci function
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Function Behavior

With the help of graphs and tables, determine a variety of function behaviors.

Find intercepts of functions (CCSS.Math.Content.HSF-IF.B.4):

y intercept of f(x) = -3x + 2
x intercept of f(x) = x^2 - 8x + 7
intercepts of -x^2 + 2x + 3
zeros of y = x^2 + 7x + 10
roots of 10x^3 + 18x^2 - 52x - 48
intercepts of y = 4^(x + 1)
x intercepts of y = (x + 4) (x - 3)/(2 x + 5)
what is the y intercept of y = x^3 - 7?

Identify and locate maximum and minimum values of functions (CCSS.Math.Content.HSF-IF.B.4):

y = x^2 + 2x - 4 minimum value
max of -2x^2 + 4x - 1
4x^3 + 3x^2 local min and max
maximum of 5x^3 - 15x on (-2, 1)

Describe end behavior of a function (CCSS.Math.Content.HSF-IF.B.4):

horizontal asymptote of f(x)= 3^(-x) + 4
(x^2 + 6x - 8)/(2x^2 + 4x + 2) horizontal asymptote
x^3/(5x^2 + 10x + 2) slant asymptote

Determine intervals on which a function has a given behavior (CCSS.Math.Content.HSF-IF.B.4):

where is f(x) = 3x^2 + 9x increasing?
3x^3 - 5x^2 - 8 decreasing
x^2 + x - 20 < 0
where is 5x (x + 7) (x - 4) (x - 8) (x - 14) increasing or decreasing?

Find the period of a function (CCSS.Math.Content.HSF-IF.B.4):

period of sin(4x) - 2
period of 5 cos(0.5(x - 1))
period of tan(x)
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Computing with Functions

Investigate functions by interpreting and manipulating their equations.

Calculate the average rate of change of a function (CCSS.Math.Content.HSF-IF.B.6):

average rate of change
(3^(2) - 3^(0))/(2 - 0)

Factor and solve quadratic functions (CCSS.Math.Content.HSF-IF.C.8a):

factor x^2 + 5x + 6
3x^2 + 10x + 8 factored form
solve x^2 + 10x + 16 by factoring
complete the square for x^2 - 12x + 23
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Graphing Functions

Use a function's equation and knowledge of key features of a function to sketch its graph.

Graph linear and quadratic functions (CCSS.Math.Content.HSF-IF.C.7a):

graph y = 4x + 2
draw the graph of y - 2 = -0.5 (x - 1)
graph line with slope = 1/3 and y-intercept = 1
draw the line through (5, 6) and (-2, -8)
plot y = -x^2 + 1
parabola through (4, 1), (-1, -4), (-6, 1)
y = x^2 + 2x - 3 axis of symmetry

Graph square root, cube root and piecewise functions (CCSS.Math.Content.HSF-IF.C.7b):

graph y = 5 * sqrt(x) + 4
plot x^(1/3)
f(x) = 4 * the cube root of x
y = |x|
Piecewise[{{x^2, x < 0}, {x, x > 0}}]
Piecewise[{{-3x, x<2}, {x + 1, 2<x<=5}, {6, 5<x<8}}]

Graph polynomial functions (CCSS.Math.Content.HSF-IF.C.7c):

plot x^3 - 6x^2 + 4x + 12
graph y = x^5 + 5x^4 + x^3 - 21x^2 - 18x
draw the graph of (x - 2) (x + 4) (x +1)^2
show the zeros of x^3 - 5 x^2 - 17 x + 21

Graph rational functions (CCSS.Math.Content.HSF-IF.C.7d):

draw the graph of (x^2 - 7x - 18)/(x^2 + 7x + 10)
plot 1/x+2
discontinuities of (x^2 + 5x + 6)/(x^2 + 7x + 12)
where is (x^2 - 4)/(x^2 - 2x - 3) discontinuous?
(2x^2 + 10x + 8)/(x^2 - 1) asymptotes

Graph exponential, logarithmic and trigonometric functions (CCSS.Math.Content.HSF-IF.C.7e):

plot f(x) = 5 * 2^(3x)
graph y = 2^x and y = 4^x
show the y-intercept of f(x) = 3 (4^x) - 2
graph log10(x)
plot y = 3 * log_2(x)
graph f(x) = 2 sin(4x) - 1
graph cos(x) and 3cos(x + pi/2)
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