10.2: Vertical Shifts of Quadratic Functions
Suppose the marketing department of a company is designing a new logo that includes a parabola. They've drawn the logo on a piece of graph paper, but they've decided that they want to change the position of the parabola by moving it 3 units down. If the original equation of the parabola was
Watch This
Multimedia Link: For more information regarding stopping distance, watch this: CK12 Basic Algebra: Algebra Applications: Quadratic Functions
 YouTube video.
Guidance
Compare the five parabolas. What do you notice?
The five different parabolas are congruent with different
The equation
The vertical movement along a parabola’s line of symmetry is called a vertical shift.
Example A
Determine the direction, shape, and
Solution: The value of
 Because
a is positive, the parabola opens upward.  Because
a is greater than 1, the parabola is narrow about its line of symmetry.  The value of
c is –4, so they− intercept is (0, –4).
Projectiles are often described by quadratic equations. When an object is dropped from a tall building or cliff, it does not travel at a constant speed. The longer it travels, the faster it goes. Galileo described this relationship between distance fallen and time. It is known as his kinematical law. It states the “distance traveled varies directly with the square of time.” As an algebraic equation, this law is:
Use this information to graph the distance an object travels during the first six seconds.



0  0 
1  16 
2  64 
3  144 
4  256 
5  400 
6  576 
The parabola opens upward, and its vertex is located at the origin. Since
Example B
Anne is playing golf. On the fourth tee, she hits a slow shot down the level fairway. The ball follows a parabolic path described by the equation,
Describe the shape of this parabola. What is its
Solution: The value of
 Because
a is negative, the parabola opens downward.  Because
a is between –1 and 1, the parabola is wide about its line of symmetry.  The value of
c is 0, so they− intercept is (0, 0).
The distance it takes a car to stop (in feet) given its speed (in miles per hour) is given by the function
Graph the function by making a table of speed values.



0  0 
10  15 
20  40 
30  75 
40  120 
50  175 
60  240 
 The parabola opens upward with a vertex at (0, 0).
 The line of symmetry is
x=0 .  The parabola is wide about its line of symmetry.
Using the function to find the stopping distance of a car travelling 65 miles per hour yields:
The Affect of Vertical Shifting on the
Consider the graphs of quadratic functions in the beginning of this Concept:
The graph of
Example C
Find the
Solution:
We can see that the \begin{align*}x\end{align*}
To factor \begin{align*}y=x^21\end{align*}
\begin{align*}y=x^21=(x1)(x+1)\end{align*}
Since \begin{align*}y=0\end{align*}
\begin{align*}0=y=(x1)(x+1)\Rightarrow (x1)=0 \text{ or }(x+1)=0 \Rightarrow x=1 \text{ or }x=1 .\end{align*}
So, the \begin{align*}x\end{align*}
Guided Practice
Determine the direction, shape, \begin{align*}y\end{align*}
Solution:
Since \begin{align*}a=1\end{align*}
\begin{align*}y=x^23x+18=(0)^23(0)+18=18\end{align*}
So the \begin{align*}y\end{align*}
To find the \begin{align*}x\end{align*}
\begin{align*}y=x^23x+18=[x^2+3x18]=[x^23x+6x18]=[x(x3)+6(x3)]=[(x+6)(x3)]\end{align*}
This means that \begin{align*}y=0\end{align*}
\begin{align*}x+6=0 \text{ or } x3=0 \Rightarrow x=6 \text{ or } x=3.\end{align*}
Thus, the \begin{align*}x\end{align*}
Practice
Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both. CK12 Basic Algebra: Graphs of Quadratic Functions (16:05)
 Using the parabola below, identify the following:
 Vertex

\begin{align*}y\end{align*}
y− intercept 
\begin{align*}x\end{align*}
x− intercepts  Domain
 Range
 Line of symmetry
 Is \begin{align*}a\end{align*}
a positive or negative?  Is \begin{align*}a\end{align*}
a \begin{align*}1<a<1\end{align*}−1<a<1 or \begin{align*}a<1\end{align*}a<−1 or \begin{align*}a>1\end{align*}a>1 ?
 Use the stopping distance function from the Concept to find:

\begin{align*}d(45)\end{align*}
d(45)  What speed has a stopping distance of about 96 feet?

\begin{align*}d(45)\end{align*}
 Using Galileo’s Law from the Concept, find:
 The distance an object has fallen at 3.5 seconds
 The total distance the object has fallen in 3.5 seconds
Which has a more positive \begin{align*}y\end{align*}

\begin{align*}y=x^2\end{align*}
y=x2 or \begin{align*}y=4x^2\end{align*}y=4x2 
\begin{align*}y=2x^2+4\end{align*}
y=2x2+4 or \begin{align*}y=\frac{1}{2} x^2+4\end{align*}y=12x2+4 
\begin{align*}y=2x^22\end{align*}
y=−2x2−2 or \begin{align*}y=x^22\end{align*}y=−x2−2
Identify the vertex and \begin{align*}y\end{align*}

\begin{align*}y=x^22x8\end{align*}
y=x2−2x−8 
\begin{align*}y=x^2+10x21\end{align*}
y=−x2+10x−21 
\begin{align*}y=2x^2+6x+4\end{align*}
y=2x2+6x+4
Which equation has a larger vertex?

\begin{align*}y=x^2\end{align*}
y=x2 or \begin{align*}y=4x^2\end{align*}y=4x2 
\begin{align*}y=2x^2\end{align*}
y=−2x2 or \begin{align*}y=2x^2 2\end{align*}y=−2x2−2 
\begin{align*}y=3x^23\end{align*}
y=3x2−3 or \begin{align*}y=3x^26\end{align*}y=3x2−6
 Nadia is throwing a ball to Peter. Peter does not catch the ball and it hits the ground. The graph shows the path of the ball as it flies through the air. The equation that describes the path of the ball is \begin{align*}y=4+2x0.16x^2\end{align*}
y=4+2x−0.16x2 . Here, \begin{align*}y\end{align*}y is the height of the ball and \begin{align*}x\end{align*}x is the horizontal distance from Nadia. Both distances are measured in feet. How far from Nadia does the ball hit the ground? At what distance, \begin{align*}x\end{align*}x , from Nadia, does the ball attain its maximum height? What is the maximum height?  Peter wants to enclose a vegetable patch with 120 feet of fencing. He wants to put the vegetable patch against an existing wall, so he needs fence for only three of the sides. The equation for the area is given by \begin{align*}a=120 xx^2\end{align*}
a=120x−x2 . From the graph, find what dimensions of the rectangle would give him the greatest area.
Mixed Review
 Factor \begin{align*}6u^2 v11u^2 v^210u^2 v^3\end{align*}
6u2v−11u2v2−10u2v3 using its GCF.  Factor into primes: \begin{align*}3x^2+11x+10\end{align*}
3x2+11x+10 .  Simplify \begin{align*} \frac{1}{9} (63) \left( \frac{3}{7} \right)\end{align*}.
 Solve for \begin{align*}b: b+2=9\end{align*}.
 Simplify \begin{align*}(4x^3 y^2 z)^3\end{align*}.
 What is the slope and \begin{align*}y\end{align*}intercept of \begin{align*}7x+4y=9?\end{align*}
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Here you'll learn about vertical shifts and their effects on the anatomy of a parabola.