Parent Function, Domain, and Range:
- Equation: y = |x|
- Domain: All real numbers
- Range: All real numbers greater than or equal to 0. (y ≥ 0)
- y-intercept: (0,0)
- x-intercept: (0,0)
- Line of symmetry: (x = 0)
- Vertex: (0,0)
Standard Form and its uses:
The absolute value of a real number x is defined by the following:
|x| = x if x ≥ 0 -x if x ≤ 0 If n is a positive number, there are two solutions to the equation |f (x)| = n because there are exactly two numbers with the absolute value equal to n: n and -n. The existence of two distinct solutions is clear when the equation is solved graphically. An absolute value function can be presented as y = a|x - h|+ k. The graph moves as the changes of slope a, x-intercept h, and y-intercept k. · Graphs of absolute value functions look like an angle (letter V) · The values of a, h, and k alters the shape of the V. · Vertex (h, k) |
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Vertex Form and its uses:
The graph of the absolute value function f (x) = | x| is similar to the graph of f (x) = x. The vertex form is intended to find the vertex of the graph and whether is faces upward or downward, and the width or narrowness of the v, also the corresponding points.
Consider the function: g(x) = x +1 + 3
Vertex =(x,y)
Look at the absolute value section: x +1
What would x have to be to get the value of zero inside the absolute value?
Answer: x = –1 (this is the value of x for the vertex)
What is being added or subtract from the absolute value group?
Answer: + 3 (this is the value of y for the vertex)
The vertex of the above function is ( –1 , 3 ) !!!
|x| =x, if x > 0
0, if x = 0
-x, if x < 0
The graph of this piece wise function consists of two rays, is V-shaped, and opens up.
The corner point of the graph, called the occurs at the origin.
Notice that the graph of y = |x| is symmetric in the y-axis because for every point
(x, y) on the graph, the point (-x, y) is also on the graph.
Consider the function: g(x) = x +1 + 3
Vertex =(x,y)
Look at the absolute value section: x +1
What would x have to be to get the value of zero inside the absolute value?
Answer: x = –1 (this is the value of x for the vertex)
What is being added or subtract from the absolute value group?
Answer: + 3 (this is the value of y for the vertex)
The vertex of the above function is ( –1 , 3 ) !!!
|x| =x, if x > 0
0, if x = 0
-x, if x < 0
The graph of this piece wise function consists of two rays, is V-shaped, and opens up.
The corner point of the graph, called the occurs at the origin.
Notice that the graph of y = |x| is symmetric in the y-axis because for every point
(x, y) on the graph, the point (-x, y) is also on the graph.
Types of Translations:
Translation: the shifting of a graph vertically, horizontally, or both;
horizontal shift = left/right shift
vertical shift = up/down shift
diagonal shift = a combination of horizontal and vertical shifts
horizontal shift = left/right shift
vertical shift = up/down shift
diagonal shift = a combination of horizontal and vertical shifts
Hmmm. The narrowed “V” is upside down! It is a reflection of f(x) = 2|x|. This means it is compressed by 2 units and reflected across the x-axis. How exciting!
The domain doesn’t change but the range is now all real numbers ≤ 0.
The domain doesn’t change but the range is now all real numbers ≤ 0.
Notable Points:
Axis of symmetry: the vertical line that goes through the vertex and
cuts the graph "in half"; it's the x coordinate of the vertex Vertex: the minimum or maximum point on an absolute value graph; it is an ordered pair, so to find the xcoordinate on an absolute value graph, set the stuff inside | |= 0 and solve to find the ycoordinate, substitute the x value into the equation @: (-h,k) Steps for graphing an absolute value function: 1. Calculate the vertex (and axis of symmetry) 2. Create an x/y table by putting the vertex in the "middle" 3. Calculate at least 5 xvalues and yvalues 4. Plot the points from your table and connect them to form your "V" shaped graph. |
How to Graph:
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