Find the coordinates of the midpoint M(x, y) on the line segment joiningP(−3,8)andQ(7,−1).
Solution
M=(−3+72,8+(−1)2)=(2,72)
2.2 Graphs of Equations
The graph of an equation in two variables, such as x and y, is the set of all ordered pairs (a, b) in the coordinate plane that satisfies the equation. A graph of an equation, then, is a picture of its solution set.
Sketching the graph of an equation by plotting points
Step 1 Make a representative table of solutions of the equation.
Step 2 Plot the solutions as ordered pairs in the coordinate plane.
Step 3 Connect the solutions in Step 2 with a smooth curve.
Intercepts
The x-intercept is the x-coordinate of a point on the graph where the graph touches or crosses the x-axis. To find the x-intercepts, set y=0 in the equation and solve for x.
The y-intercept is the y-coordinate of a point on the graph where the graph touches or crosses the y-axis. To find the y-intercepts, set x=0 in the equation and solve for y.
Symmetry A graph is symmetric with respect to:
the y-axis if for every point (x, y) on the graph, the point (−x,y) is also on the graph. That is, replacing x with −x in the equation produces an equivalent equation.
the x-axis if for every point (x, y) on the graph, the point (x,−y) is also on the graph. That is, replacing y with −y in the equation produces an equivalent equation.
the origin if for every point (x, y) on the graph, the point (−x,−y) is also on the graph. That is, replacing x with −x and y with −y in the equation produces an equivalent equation.
Circle A circle is the set of points in a Cartesian coordinate plane that are at a fixed distance r from a specified point (h, k). The fixed distance r is called the radius of the circle, and the fixed point (h, k) is called the center of the circle. The equation (x−h)2+(y−k)2=r2 is called the standard form of an equation of a circle.
The general form of an equation of a circle is
x2+y2+ax+by+c=0.
Refer to Example1 on page 186 to review sketching the graph of an equation by plotting points.
Find the x- and y-intercepts and identify the symmetries for the graph of the equationx2+y2=25,shown below.
Solution
The graph is symmetric with respect to the x-axis, the y-axis, and the origin. Both the x- and y-intercepts are −5 and 5.
Write the equation of a circle:
x2+y2+4x−6y−3=0
in standard form and give the center and radius of the circle.
Solution
Complete the square by adding both 4 and 9 to both sides of the given equation.
Any set of ordered pairs is a relation. The set of all first components is the domain, and the set of all second components is the range. The graph of all of the ordered pairs is the graph of the relation.
A function is a relation in which each element of the domain corresponds to exactly one element in the range.
If a function f is defined by an equation, then its domain is the largest set of real numbers for which f(x) is a real number.
Vertical-line test. If each vertical line intersects a graph at no more than one point, the graph is the graph of a function.
Shown below is the graph of the relation y=x3, which has the set of all real numbers as both domain and range. The relation y=x3 defines a function because each element of the domain corresponds to exactly one element in the range.
Notice that each vertical line intersects the graph of y=x3 at no more than one point.
The graph of y2=−8x shown below does not pass the vertical-line test, so it is not the graph of a function.
2.5 Properties of Functions
Let x1 and x2 be any two numbers in the interval (a, b).
f is increasing on (a, b) if x1<x2 implies f(x1)<f(x2).
f is decreasing on (a, b) if x1>x2 implies f(x1)>f(x2).
f is constant on (a, b) if x1<x2 implies f(x1)=f(x2).
Assume (x1,x2) is an interval in the domain of f containing a number a.
f(a) is relative minimum of f if f(a)≤f(x) for every x in (x1,x2).
f(a) is relative maximum of f if f(a)≥f(x) for every x in (x1,x2).
A point (a, b) on the graph of f is a turning point if the graph of f changes direction at (a, b) from increasing to decreasing or from decreasing to increasing.
Suppose that for each x in the domain of f, −x is also in the domain of f.
f is called an even function if f(−x)=f(x). The graph of an even function is symmetric about the y-axis.
f is called an odd function if f(−x)=−f(x). The graph of an odd function is symmetric about the origin.
The average rate of change of f(x) as x changes from a to b is defined by f(b)−f(a)b−a,b≠a.
The expression f(x)−f(a)x−a,x≠a, is called a difference quotient at a. The expression f(x+h)−f(h)h,h≠0, is also called a difference quotient.
The graph of f(x)=x2 is decreasing on (−∞,0) and increasing on (0,∞),f(0)=0 is a relative minimum of f, and (0, 0) is turning point. Also, because f(−x)=(−x)2=x2,f(x)=f(−x) and f is an even function.
Find the average rate of change off(x)=x2as x changes from −3 to 1.
Solution
The average rate of change is:
f(1)−f(−3)1−(−3)=1−94=−2.
2.6 A Library of Functions
Basic Functions
Identity function
f(x)=x
Constant function
f(x)=c
Squaring function
f(x)=x2
Cubing function
f(x)=x3
Absolute value function
f(x)=|x|
Square root function
f(x)=x−−√
Cube root function
f(x)=x−−√3
Reciprocal function
f(x)=1x
Reciprocal square function
f(x)=1x2
Greatest integer function
f(x)=〚x〛
For piecewise functions, different rules are used in different parts of the domain. See page 253.
Refer to the Library of Functions in Section2.6. The graphs of the basic functions can also be found on the final pages just inside the back cover of the text.
2.7 Transformations of Functions
Vertical and horizontal shifts. Let f be a function and c and d be positive numbers.
To graph
Shift the graph of f
f(x)+d
up d units
f(x)−d
down d units
f(x−c)
right c units
f(x+c)
left c units
Reflections:
To graph
Reflect the graph of f about the
−f(x)
x-axis
f(−x)
y-axis
Stretching and compressing. The graph of g(x)=af(x) for a>0 has the same shape as the graph of f(x) and is stretched vertically away from the x-axis if a>1 and compressed vertically toward the x-axis if 0<a<1. If a is negative, the graph of |a|f(x) is reflected about the x-axis.
The graph of g(x)=f(bx) for b>0 is obtained from the graph of f by stretching it away from the y-axis if 0<b<1 and compressing it horizontally toward the y-axis if b>1. If b is negative, the graph of f(|b|x) is reflected about the y-axis.
Stretch the graph off(x)=−3−x−−−√−1.
Solution
The graph of f(x)=−3−x−−−√−1 is the graph of f(x)=x−−√ reflected about the y-axis, then reflected about the x-axis, stretched vertically by a factor of 3, and then shifted 1 unit down.
2.8 Combining Functions; Composite Functions
Given two functions f and g, for all values of x for which both f(x) and g(x) are defined, the functions can be combined to form sum, difference, product, and quotient functions.
The composition of f and g is defined by (f∘g)(x)=f(g(x)).
The input of f is the output of g. The domain of f∘g is the set of all x in the domain of g such that g(x) is in the domain of f.
In general, f∘g≠g∘f.
Letf(x)=x2−5x+4 and g(x)=x−4.
Findf+g,f−g,f⋅g,andfgand the domain of each function.
A function f is one-to-one if for any x1 and x2 in the domain of f, f(x1)=f(x2) implies x1=x2.
Horizontal-line test. If each horizontal line intersects the graph of a function f in at most one point, then f is a one-to-one function.
Inverse function. Let f be a one-to-one function. Then g is the inverse of f, and we write g=f−1 if (f∘g)(x)=x for every x in the domain of g and (g∘f)(x)=x for every x in the domain of f. The graph of f−1 is the reflection of the graph of f about the line y=x.
If f is a one-to-one function, we find f−1 by the four-step procedure on page 299.
The function f(x)=x2 is not a one-to-one function because, for example, y=4 corresponds to both x=2 and x=−2. That is, f(2)=f(−2) but 2≠−2.
Determine thatf(x)=3x−6is a one-to-one function. Find.
Solution
The graph of f(x)=3x−6 is a straight line with slope 3. By the horizontal-line test f is a one-to-one function.