Name
MATH 154 PRACTICE MIDTERM III
Please work out each of the given problems. Credit will be based on the steps
that you show towards the final answer. Do all your work and give all your
answers on you own sheet of paper. Show your work.
Problem 1 (20 Po
Nonlinear Systems
To solve a system of two equations and two unknowns when the equations are not linear, we use
the methods of substitution or elimination and hope that the resulting equation becomes a hidden
quadratic or other solvable equation.
Example:
MATH 154 Practice Midterm I Key
Please work out each of the given problems. Credit will be based on the steps that you
show towards the final answer. Show your work.
Problem 1 Solve the inequality. Write the solution on a number line.
2x2  5x > 3
Solutio
Name
MATH 154 PRACTICE MIDTERM II
Please work out each of the given problems. Credit will be based on the steps
that you show towards the final answer. Do all your work and give all your
answers on you own sheet of paper. Show your work.
Problem 1
Solve t
Inverses
Inverses
Definition
The inverse of a relation R is the relation consisting of all ordered
pairs (y,x) such that (x,y) belongs to R
Example:
The inverse of the relation
(2,3), (4,5), (2,6), (4,6)
is
(3,2), (5,4), (6,2), (6,4)
Generally we switch t
Exercises Involving Distance and Circles
1. The distance between points (a,b) and (c,d) is
.
3. The circle with radius r and center at the point (0,k) is given by the equation x2 + (y  k)2 =
r2.
5. If the distance between the point P and the point (3,4)
Name
MATH 154 PRACTICE FINAL
Please work out each of the given problems. Credit will be based on the steps
that you show towards the final answer. Do all of your work and show your
solutions on your own sheet of paper. Show your work.
Problem 1
Solve
Solu
More Parabolas
Horizontal Shifting
Consider the graphs
y=
1.
(x + 0)2
2.
(x + 1)2
3.
(x + 2)2
4.
(x + 3)3
We see that adding a number inside the
parenthesis shifts the graph left or
right. The rules below state this
precisely.
Rule1: f(x  a) = f(x) shift
Parabolas
Parabolas
Consider the plot of
y = x2
to the right
The point at the bottom (or top) is called the vertex and the
line that cuts the parabola into two equal pieces is called
the axis of symmetry.
Example
sketch the graph of the parabola
y = x2 
3 by 3 Linear Systems
Geometry of 3X3 systems
Recall that for lines, either they intersect in a point, are parallel, or are the same line. Similarly,
if we have three planes either they intersect in a point, a line, don't intersect at all, or are the same
Solutions to the Odd Problems on Hyperbolae Centered at
the Origin
1. For the hyperbola
x2
y2

9
= 1
4
the vertices are at the points (3,0) and (3,0) .
3. The asymptotes of the hyperbola
x2
y2

9
= 1
4
have equations y = 2/3 x and y = 2/3 x .
5. The g
Answers to Odd Exercises on Ellipses Centered at the Origin
1. For the ellipse
x2
y2
+
9
= 1
4
the xaxis the MAJOR axis and the yaxis is the MINOR axis.
3. Ellipses are never graphs of functions.
True, since they will never pass the vertical line test.
Discriminant, Applications, Hidden Quadratics
The Discriminant and Factoring
If we have a quadratic expression
ax2 + bx + c
and want to determine if it factors, then we can ask an equivalent question about whether there
are rational roots. In particular,
Circles and Distance
The Distance Formula
Recall that the Pythagorean Theorem states that if a, b, c are sides of a right triangle with c the
hypotenuse, then
a2 + b2 = c2
Let (x,y) be a point in the plane. Then if we draw the triangle with one vertex at
Nonlinear Inequalities and the Pythagorean Theorem
Quadratic Inequalities
We will solve using the following steps
1.
Put everything on the left hand side so that we have for example Quad > 0.
2.
Factor and set equal to zero.
3.
Solve and place answers on
Relations and Functions
Relations
A relation is a rule that takes an input from a set (called the domain) and gives one or
more outputs of another set (called the range).
Examples
1.
(0,0), (4,4), (0,3), (2,1)
2.
y = 2x
3.
4. The circle is a relation. We
Ellipses and Hyperbolae
Draw an Ellipse With a String and Two Fixed Points
Geometrically an ellipse is defined as follows: Let P and Q be fixed points in the plane and let k
be a positive real number. Then a point R is in the ellipse if the sum of the dis