graphs of functions digital lesson. copyright © by houghton mifflin company, inc. all rights...
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Graphs of Functions
Digital Lesson
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x
y
4
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The domain of the function y = f (x) is the set of values of x for which a corresponding value of y exists.
The range of the function y = f (x) is the set of values of y which correspond to the values of x in the domain.
Domain
Range
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x
y
– 1
1
Example: Find the domain and range of the function f (x) = from its graph.
The domain is [–3,∞).
The range is [0,∞).
3x
Range
Domain
(–3, 0)
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x
y
4
-4
Vertical Line Test
A relation is a function if no vertical line intersects its graph in more than one point.
This graph does not pass the vertical line test. It is not a function.
This graph passes the vertical line test. It is a function.
y = x – 1x = | y – 2|
x
y
4
-4
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• decreasing on an interval if, for any x1 and x2 in the interval, x1 < x2 implies f (x1) > f (x2),
• constant on an interval if, for any x1 and x2 in the interval, f (x1) = f (x2).
The graph of y = f (x):
• increases on (– ∞, –3),
• decreases on (–3, 3),
• increases on (3, ∞).
A function f is:
• increasing on an interval if, for any x1 and x2 in the interval, x1 < x2 implies f (x1) < f (x2),
(3, – 4)
x
y(–3, 6)
–2
2
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Increasing/Decreasing Intervals
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Calculator Mode Screen
Keep your calculator in degree mode until we get to trigonometry.
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Standard Window ScreenThis is where you change
the size of your graph.Xmin is the leftmost point on
your graph.Xmax is the rightmost point
on your graph.Xscl is the scale you are using
on your x-axis.Ymin is the lowest point on
your graph.Ymax is the highest point on
your graph.Yscl is the scale that you are
using on your y-axis.
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The Math Menu
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Full, Horizontal, G-T Options
This alteration is made in the MODE options.
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Grids
FORMAT is located in the text above the ZOOM button located at the top center of your calculator.
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A function value f(a) is called a relative minimum of f if there is an interval (x1, x2) that contains a such that
x1 < x < x2 implies f(a) f(x).
x
y
A function value f(a) is called a relative maximum of f if there is an interval (x1, x2) that contains a such that
x1 < x < x2 implies f(a) f(x).
Relative minimum
Relative maximum
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Relative Max and Min
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Relative Max/Min
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Graphing Utility: Approximate the relative minimum of the function f(x) = 3x2 – 2x – 1.
– 6
– 6
6
6
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x
y
4
-4
A piecewise-defined function is composed of two or more functions.
f(x) =3 + x, x < 0
x2 + 1, x 0 Use when the value of x is less than 0.
Use when the value of x is greater or equal to 0.
(0 is not included.)
open circle
(0 is included.)
closed circle
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Piecewise Functions
TEST is located in the text right above the MATH button on the left side of the calculator.
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Piecewise Functions
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Piecewise Functions
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Greatest Integer Function
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Greatest Integer Functions
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A function f is even if for each x in the domain of f, f (– x) = f (x).
x
yf (x) = x2
f (– x) = (– x)2 = x2 = f (x)
f (x) = x2 is an even function.
Symmetric with respect to the y-axis.
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A function f is odd if for each x in the domain of f, f (– x) = – f (x).
x
y
f (x) = x3
f (– x) = (– x)3 = –x3 = – f (x)
f (x) = x3 is an odd function.
Symmetric with respect to the origin.
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Even/Odd Functions
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Even/Odd Functions
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Even/Odd Function
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