Question

name _____________ date _____________
more rabbit problems
when you sketch the graphs, include the x-intercepts and vertex.

length of fencing

function

vertex

axis of symmetry

sketch graph

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50 ft

$a(w)=w(25-w)$

$(12.5, 156.25)$

$x=12.5$

sketched graph

164 ft

$a(w)=w(82-w)$

$(41, 1,681)$

$x=41$

sketched graph

66 ft

30 ft

$a(w)=w(15-w)$

$x=50$

$a(w)=w(35-w)$

$a(w)=w(17-w)$

$(8.5, 72.25)$

$x=8.5$

sketched graph with intercepts (0,0), (17,0) and vertex (8.5,72.25)

focus on algebra: quadratic functions 2014

w-54

Explanation:

Step1: 66 ft row: Define area function

For fencing length $L=66$ ft, the area function for a rectangular enclosure (where one side uses existing fencing, so $L = 2w + l$, $l = 33 - w$) is:
$A(w) = w(33 - w)$

Step2: 66 ft row: Find vertex x-coordinate

Axis of symmetry (vertex x-value) for $A(w)=-w^2+33w$ is $w=-\frac{b}{2a}=-\frac{33}{2(-1)}=16.5$

Step3: 66 ft row: Find vertex y-coordinate

Substitute $w=16.5$ into $A(w)$:
$A(16.5)=16.5(33-16.5)=16.5^2=272.25$
Vertex: $(16.5, 272.25)$
Axis of symmetry: $x=16.5$
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Step4: 30 ft row: Identify fencing length

From $A(w)=w(15-w)$, $l=15-w$, so $L=2w+l=2w+15-w=15+w$; intercepts at $w=0,15$, so total fencing $L=30$ ft (matches table)

Step5: 30 ft row: Find vertex x-coordinate

Axis of symmetry: $w=-\frac{b}{2a}=-\frac{15}{2(-1)}=7.5$

Step6: 30 ft row: Find vertex y-coordinate

$A(7.5)=7.5(15-7.5)=7.5^2=56.25$
Vertex: $(7.5, 56.25)$
Axis of symmetry: $x=7.5$
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Step7: $x=50$ row: Find fencing length

Axis of symmetry $w=50$, so $w=-\frac{b}{2a}=50$. For $A(w)=w(L/2 - w)$, $L/2=100$, so $L=200$ ft

Step8: $x=50$ row: Define area function

$A(w)=w(100 - w)$

Step9: $x=50$ row: Find vertex y-coordinate

$A(50)=50(100-50)=2500$
Vertex: $(50, 2500)$
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Step10: $A(w)=w(35-w)$ row: Find fencing length

$l=35-w$, so $L=2w+l=2w+35-w=35+w$; intercepts at $w=0,35$, total fencing $L=70$ ft

Step11: $A(w)=w(35-w)$ row: Find vertex x-coordinate

Axis of symmetry: $w=-\frac{b}{2a}=-\frac{35}{2(-1)}=17.5$

Step12: $A(w)=w(35-w)$ row: Find vertex y-coordinate

$A(17.5)=17.5(35-17.5)=17.5^2=306.25$
Vertex: $(17.5, 306.25)$
Axis of symmetry: $x=17.5$

Answer:

Length of Fencing	Function	Vertex	Axis of Symmetry	Sketch Graph (Key Points)
30 ft	$A(w)=w(15-w)$	$(7.5, 56.25)$	$x=7.5$	x-intercepts $(0,0),(15,0)$, vertex $(7.5, 56.25)$
200 ft	$A(w)=w(100-w)$	$(50, 2500)$	$x=50$	x-intercepts $(0,0),(100,0)$, vertex $(50, 2500)$
70 ft	$A(w)=w(35-w)$	$(17.5, 306.25)$	$x=17.5$	x-intercepts $(0,0),(35,0)$, vertex $(17.5, 306.25)$