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PreCalculus Mathematics Quiz #4 Sample Solutions
1. Give the slopeintercept equation of the line passing
through the points (1,−3) and (5, 3).
In the space below, sketch the line.
Solution: First we ascertain the line’s slope:
Having the line’s slope, as well as two of its points (we
now need only one of them), we can use
the pointslope equation form in order to derive the slopeintercept
equation:
(point − slope form )  
(plugging in 5, 3, and for x_{1}, y_{1},m)  
(arithmetic, algebra )  
(arithmetic) 
As an alternative to using the pointslope equation (as we
did above), we could have recognized
that, for some b yet to be determined, the line we seek is described by the
equation ,
and, in particular, this equation is satisfied by taking x and y to be x_{1} and
y_{1}, respectively, for
any point (x_{1}, y_{1}) on the line.
(plugging in 5, 3 for x_{1}, y_{1})  
(arithmetic)  
(algebra)  
(arithmetic) 
Having solved for b , we get that the pointslope form is
, which agrees with the
solution we obtained before. See graph in Figure 1.
2. Give the slopeintercept equation of the line that
passes through the point (−2, 3) and is
perpendicular to the line given by the equation x − 2y = 4. Sketch both lines in
the space
below.
Solution: Subtracting 4 from both sides of the given equation yields x −
2y − 4 = 0, which is
of the form Ax + By + C = 0. Recall that the slope of the corresponding line is
, which in
this case (with A = 1 and B = −2) is .
Had you not remembered this technique, you could have taken the given equation
and, by
subtracting 4 from each side , adding 2y to each side, and then dividing each
side by 2, obtained
Figure 1: Answer to (1)
the equivalent equation . This, of course, is
in the slope intercept format and reveals
that the line has a slope of
.
Having ascertained that
is the slope of a line that is perpendicular to the line that we seek,
we conclude that the line that we seek has slope −2. (After all, the product of
the slopes of
any two perpendicular lines (neither of which is vertical) is −1, and
.)
In addition to knowing one of its points, namely (−2, 3), now we also know the
slope of the line
that we seek, namely −2. Beginning with the pointslope equation, we derive the
slopeintercept
equation:
(point − slope form)  
(plugging in − 2, 3,−2 for x_{1}, y_{1},m)  
(arithmetic)  
(arithmetic, algebra)  
(arithmetic) 
Figure 2: Answer to (2)
3. In the space below, sketch the parabola described by
the equation y = x^{2} + 2x + 3. Label
the vertex by its coordinates .
Solution: We use the completingthesquare method to transform the given
equation into one
of the form y = a(x − h)^{2} + k, which describes a parabola with vertex (h, k).
(given)  
(add 1 and subtract 1)  
( factoring and arithmetic ) 
We conclude that the parabola has vertex (−1, 2) and opens
upward. Indeed, the graph is just
that of y = x^{2} shifted one unit to the left and two units up.
Figure 3: Answer to (3)
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