3.2.1: Exercises 3.2
- Page ID
- 63464
Terms and Concepts
Exercise \(\PageIndex{1}\)
In which situations is substitution a more appropriate solution method than equating the functions?
- Answer
-
When one or both functions are defined implicitly.
Exercise \(\PageIndex{2}\)
Is \(y=x^3+5x-7\) an implicitly or explicitly defined function? Explain.
- Answer
-
Explicitly; \(y\) is isolated.
Exercise \(\PageIndex{3}\)
Is \(xy+y^2 -y = 2x+6\) an implicitly or explicitly defined function? Explain.
- Answer
-
Implicitly; \(y\) is not isolated.
Exercise \(\PageIndex{4}\)
Describe the pros and cons of using graphing to find the point(s) of intersection.
- Answer
-
Answers will vary, but graphing helps you determine how many intersections points exist, but does not always clearly show the exact values.
Problems
In exercises \(\PageIndex{5}\) - \(\PageIndex{8}\), determine the maximum possible number of intersections for the described functions.
Exercise \(\PageIndex{5}\)
Two linear functions with different slopes
- Answer
-
1
Exercise \(\PageIndex{6}\)
A linear function and a quadratic function
- Answer
-
2
Exercise \(\PageIndex{7}\)
Two explicitly defined quadratic functions
- Answer
-
2
Exercise \(\PageIndex{8}\)
A cubic function and a constant function
- Answer
-
3
In exercises \(\PageIndex{9}\) - \(\PageIndex{12}\), determine the minimum possible number of intersections for the described functions.
Exercise \(\PageIndex{9}\)
Two linear functions with different slopes
- Answer
-
1
Exercise \(\PageIndex{10}\)
A linear function and a quadratic function
- Answer
-
0
Exercise \(\PageIndex{11}\)
Two explicitly defined quadratic functions
- Answer
-
0
Exercise \(\PageIndex{12}\)
A cubic function and a constant function
- Answer
-
1
In exercises \(\PageIndex{13}\) - \(\PageIndex{18}\), find all points of intersection between the given functions.
Exercise \(\PageIndex{13}\)
\(y=x^2-1\) and \(y=x-1\)
- Answer
-
\((0,-1)\) and \((1,0)\)
Exercise \(\PageIndex{14}\)
\(x^2+y^2=1\) and \(4y=3x\)
- Answer
-
\((\frac{4}{5},\frac{3}{5})\) and \((-\frac{4}{5}, -\frac{3}{5})\)
Exercise \(\PageIndex{15}\)
\(y-1 = \sqrt{3x}\) and \(y=x+1\)
- Answer
-
\((0,1)\) and \((3,4)\)
Exercise \(\PageIndex{16}\)
\(y=x^2-3x+2\) and the x-axis
- Answer
-
\((1,0)\) and \((2,0)\)
Exercise \(\PageIndex{17}\)
\(y=x^2-3x+2\) and \(y=5\)
- Answer
-
\((\frac{3+\sqrt{21}}{2},5)\) and \((\frac{3-\sqrt{21}}{2},5)\)
Exercise \(\PageIndex{18}\)
\(y+2x=5\) and \(y+3=x^3-7x^2+12x\)
- Answer
-
\((1,3)\), \((2,1)\), and \((4,-3)\)
In exercises \(\PageIndex{19}\) - \(\PageIndex{22}\), sketch the region bounded by the given functions and determine all intersection points.
Exercise \(\PageIndex{19}\)
\(y=x^2\) and \(y=x\)
- Answer
-
Points of intersection are \((0,0)\) and \((1,1)\)
Exercise \(\PageIndex{20}\)
\(y=x^2\) and \(y=x+2\)
- Answer
-
Points of intersection are \((-1,1)\) and \((2,4)\)
Exercise \(\PageIndex{21}\)
\(y=x^2\) and \(y=\sqrt{x}\)
- Answer
-
Points of intersection are \((0,0)\) and \((1,1)\)
Exercise \(\PageIndex{22}\)
\(3y+2x=6\), the x-axis, and the y-axis (hint: sketch before looking for the intersection points)
- Answer
-
Points of intersection are \((0,0)\), \((0,2)\), and \((3,0)\)