Unit 7 Plan - Algebra 1

Lesson 1

A Different Kind of Change

100 Meters of Fencing

A rectangular yard is enclosed by 100 meters of fencing. The table shows some possible values for the length and width of the yard.

length (meters)	width (meters)	area (square meters)
10	40	400
20	30
25	25	625
35	15	525
40

Complete the table with the missing values.
If the values for length and area are plotted, what would the graph look like?
How is the relationship between the length and the area of the rectangle different from other kinds of relationships we’ve seen before?

Show Solution

The missing value in the second row is 600. The missing values in the last row are 10 and 400.
Sample response: The points would form a stretched out upside-down U shape, or an arch.
Sample response: As one quantity increases, the other quantity first increases and then decreases, instead of always increasing or always decreasing.

Lesson 2

How Does It Change?

Comparing Types of Growth

Here are three patterns of dots.

<p>Four steps of a growing pattern. Step 0 is blank. Step 1: 2 dots, one atop the other. Step 2: 4 dots, 2 in row 1 and 2 in row 2. Step 3: 6 dots, 3 dots on row 1 and 3 dots on row 2.</p> — Pattern A

<p>Four steps of a growing pattern.</p> — Pattern B

Which pattern shows a quadratic relationship between the step number and the number of dots? Explain or show how you know.

Show Solution

Pattern B shows a quadratic relationship. Sample explanations:

The number of dots is the step number squared plus 1, or $n^2+1$ .
(Students might answer by elimination.) Pattern A grows linearly. It is growing by 2 each time. Pattern C is growing by a factor of 3 each time, so it grows exponentially.
(Students might recall the pattern they saw in the lesson but not yet internalize the squared input.) Pattern B is growing by 1, then by 3, then by 5, which is the pattern we saw in other examples of quadratic relationships in the lesson.

Section A Check

Section A Checkpoint

Problem 1

Write an expression that shows the relationship between the step number, $n$ , and the number of dots.
Is this relationship linear, exponential, or quadratic? Explain your reasoning.

Show Solution

$n^2-1$
It is quadratic because it has a squared term in the expression for the relationship.

Lesson 3

Building Quadratic Functions from Geometric Patterns

A Quadratic Function?

Here is a pattern of squares.

<p>Three steps of a growing pattern.</p>

Write an equation to represent the relationship between the step number, $n$ , and the number of squares in the pattern. Briefly describe how each part of the equation relates to the pattern.
Is the relationship between the step number and the number of squares a quadratic function? Explain how you know.

Show Solution

$y = n^2 + n$ (or equivalent), because there is an $n$ -by- $n$ array of small squares, which gives $n^2$ , and then there is a row of $n$ squares in the lower right.
The relationship is a quadratic function because the number of squares could be expressed as $n^2+n$ , which is a quadratic expression.

Lesson 4

Comparing Quadratic and Exponential Functions

Comparing $5x^2$ and $2^x$

Tyler completes the table comparing values of the expressions $5x^2$ and $2^x$ .

$x$	$5x^2$	$2^x$
1	5	2
2	20	4
3	45	8
4	80	16

Tyler concludes that $5x^2$ will always be greater than $2^x$ for the same value of $x$ . Do you agree? Explain or show your reasoning.

Show Solution

Tyler is not correct. For small values of $x$ , $5x^2$ is greater than $2^x$ , but $2^x$ is eventually greater because it always doubles when $x$ increases by 1. When $x = 9$ , $5(9)^2 = 405$ , and $2^{9} = 512$ , so $2^x$ is larger.

Lesson 5

Building Quadratic Functions to Describe Situations (Part 1)

Where Will It Be?

The expression $16t^2$ represents the distance in feet that an object falls after $t$ seconds. The object is dropped from a height of 906 feet.

What is the height in feet of the object 2 seconds after it is dropped?
Write an expression representing the height of the object in feet $t$ seconds after it is dropped.

Show Solution

842, or $906 - 64$ (or equivalent)
$906-16t^2$

Lesson 6

Building Quadratic Functions to Describe Situations (Part 2)

Rocket in the Air

The height, $h$ , of a stomp rocket (propelled by a short blast of air) above the ground after $t$ seconds is given by the equation $h(t) = 5 + 100t - 16t^2$ . Here is a graph that represents $h$ .

<p>Graph of the quadratic function <span class="math"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>h</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mn>5</mn><mo>+</mo><mn>100</mn><mi>t</mi><mo>−</mo><mn>16</mn><msup><mi>t</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">h(t)=5+100t - 16t^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">h</span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">5</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">100</span><span class="mord mathnormal">t</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord">16</span><span class="mord"><span class="mord mathnormal">t</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></span> on a coordinate plane, origin <span class="math"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi></mrow><annotation encoding="application/x-tex">O</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span></span></span></span></span>.</p>

How does the 5 in the equation relate to the graph?
What does $100t$ in the equation mean in terms of the rocket?
What does the $\text{-}16t^2$ mean in terms of the rocket?
Estimate the time when the rocket hits the ground.

Show Solution

The graph intersects the vertical axis at 5.
It indicates that the initial velocity of the rocket was 100 feet per second upward.
This indicates the effect of gravity pulling the rocket back toward Earth.
At about 6.3 seconds.

Lesson 7

Building Quadratic Functions to Describe Situations (Part 3)

Making the Greatest Revenue

This graph represents the revenue, $r$ , in dollars that a company expects if they sell their product for $p$ dollars.

Based on this model, which price would generate more revenue for the company, $5 or $17? Explain how you know.
At what price should the company sell their product if they wish to make as much revenue as possible? How much revenue will they make?
What is an appropriate domain for the function? Explain how you know.

Show Solution

$5, because it would generate about $3,700 in revenue. Charging $17 would bring in about $2,500.
Based on the graph, a price of $10 would bring in about $5,000.
$0 \leq p \leq 20$ , because the price has to be non-negative, and the company would earn negative revenue if the price were more than $20.

Section B Check

Section B Checkpoint

Problem 1

A pattern of squares grows based on the expression $10n^2$ , where $n$ represents the step of the pattern.

A pattern of dots grows based on the expression $3^n$ , where $n$ represents the step of the pattern.

Han says that even though he doesn’t know which step it happens on, he knows that after some step, there will always be more dots than squares after that step. Do you agree or disagree? Explain your reasoning.

Show Solution

I agree. Sample reasoning: The squares grow in a quadratic pattern, and the factor of growth between each step decreases (becomes closer and closer to 1) after each step. The dots grow based on an exponential pattern, and the factor of growth is always 3.

Problem 2

A fall festival's pumpkin launcher sends pumpkins into the air so that the pumpkin's height in meters above the ground is modeled by the equation $h(t) = \text{-}5t^2 + 100t$ with $t$ as the time in seconds after launch.

A graph comparing the height above the ground in meters and the time in seconds. The x intercepts of the graph are at x=0 and x=20.

At what height above the ground was the pumpkin when it was launched?
What does the $100t$ in the equation mean in terms of the pumpkin?
About when does the pumpkin hit the ground?

Show Solution

0 meters
It means that the pumpkin was initially launched at 100 meters per second.
20 seconds after the launch

Lesson 8

Equivalent Quadratic Expressions

Writing Equivalent Expressions

Use a diagram to show that $(3x +1)(x+2)$ is equivalent to $3x^2 + 7x+2$ .
Is $(x+4)^2$ equivalent to $2x^2 + 8x + 8$ ? Explain or show your reasoning.

Show Solution

Diagram should show partial products of $3x(x)$ , $6x$ , $x$ , and 2, which add up to $3x^2 + 7x+2$ .
No. $(x+4)(x+4) = x^2 + 4x + 4x + 16 = x^2 + 8x + 16$ . (Students may use a diagram or distributive property to reason.)

Lesson 9

Standard Form and Factored Form

From One Form to Another

For each expression, write an equivalent expression in standard form. Show your reasoning.

$(2x+5)(x+1)$
$(x-2)(x+2)$

Show Solution

$2x^2 + 7x + 5$ . Sample reasoning: The diagram shows that $(2x+5)(x+1) = 2x \boldcdot x + 2x \boldcdot 1 + 5 \boldcdot x + 5\boldcdot 1$ . Adding this together gives $2x^2 + 7x + 5$ in standard form.

$2x$ $5$

$x$ $2x^2$ $5x$

$1$ $2x$ $5$
$x^2 - 4$ . Sample reasoning: $(x+2)(x-2) = x^2 + (\text-2x) +2x + (\text-4) = x^2 - 4$ . The diagram also shows that $(x+2)(x-2) = x^2-4$ .

$x$ $2$

$x$ $x^2$ $2x$

$\text-2$ $\text-2x$ $\text-4$

Section C Check

Section C Checkpoint

Problem 1

Write each equation in standard form, then identify the $x$ - and $y$ -intercepts in the graph of each.

$y = (x+1)(x-1)$
$y = (1-x)(x+3)$

Show Solution

$y = x^2 - 1$ . $x$ -intercepts: $(\text{-}1,0), (1,0)$ . $y$ -intercept: $(0,\text{-}1)$
$y = \text{-}x^2 -2x+3$ . $x$ -intercepts: $(1,0), (\text{-}3,0)$ . $y$ -intercept: $(0,3)$

Lesson 11

Graphing from the Factored Form

Sketching a Graph

Function $f$ is given by $f(x) = (x-2)(x+4)$ . Without using graphing technology, answer the following questions.

What are the $x$ -intercepts of the graph representing $f$ ?
What are the $x$ - and $y$ -coordinates of the vertex of the graph?
What is the $y$ -intercept?
Sketch a graph that represents $f$ .

Show Solution

$(2,0)$ and $(\text-4,0)$
$x = \text-1$ , $y = (\text-1-2)(\text-1+4) = \text-9$
$y$ -intercept is $(0,\text-8)$ , because $(0-2)(0+4) = \text-8$ .
The graph opens upward with vertex at $(\text-1, \text-9)$ and $x$ -intercepts at $(2,0)$ and $(\text-4,0)$ .

Lesson 12

Graphing the Standard Form (Part 1)

Matching Equations and Graphs

Here are graphs that represent three quadratic functions, defined by:

$\displaystyle f(x) = x^2 - 4\\ g(x) = 1 -x^2\\ h(x)=x^2+4$

<p>Coordinate plane with 3 quadratic functions. Curve A opens up and has minimum at 0 comma 4. Curve B opens up and has minimum at negative 4 comma 0. Curve C opens down and has maximum 0 comma 1.</p>

Match each equation to a graph that represents it. Explain how you know.
Write down the equation that can be represented by Graph C. Which part of the equation tells us that the graph opens downward?

Show Solution

Sample response:
- Graph A represents $h$ because the $y$ -intercept is $(0,4)$ , and there are no $x$ -intercepts because this function always takes positive values.
- Graph B represents $f$ because the $y$ -intercept is $(0,\text-4)$ , and the graph opens upward.
- Graph C represents $g$ because the $y$ -intercept is $(0,1)$ , and the graph opens downward.
$y=1 - x^2$ . The negative coefficient of $x^2$ makes the graph open downward.

Lesson 13

Graphing the Standard Form (Part 2)

Sketching Graphs

Consider the quadratic equation $y = x^2-4x$ . If we graph the equation, where are the $x$ -intercepts located? What is the $x$ -coordinate of the vertex?
Here is a graph of $y=x^2$ . Sketch a graph of $y = x^2 + 5x$ on the same graph. Briefly explain how you know where to sketch the graph.

Show Solution

The $x$ -intercepts are at $x = 0$ and $x = 4$ , with the vertex halfway between the intercepts, at $x = 2$ .
Sample reasoning: $x^2+5x$ is equivalent to $x(x+5)$ , so the $x$ -intercepts are at $x=0$ and $x=\text-5$ . The squared term has a positive coefficient so the graph opens upward. Adding a positive linear term to $x^2$ shifts the graph down and to the left.

Lesson 14

Graphs That Represent Situations

Beach Ball Trajectory

The equation $y=(\text-16t-2)(t-1)$ represents the height, in feet, of a beach ball thrown by a child, as a function of time, $t$ seconds after the ball is thrown.

Find the zeros of the function. Explain or show your reasoning.
What do the zeros tell us in this situation? Are both zeros meaningful?
From what height is the beach ball thrown? Explain or show your reasoning.

Show Solution

1 and $\text- \frac18$ . Sample reasoning:
- From the factored form, we can tell that the graph intersects the horizontal axis at $t=1$ and $t=\text-\frac18$ . The zeros of the function are those same values.
- (If students graph the function) The graph shows $t$ -intercepts at $(\text- \frac18,0)$ and $(1,0)$ .
The zeros mean the time in seconds when the ball hits the ground. Only one of them is meaningful. The negative zero doesn’t mean anything here.
From a height of 2 feet. Sample reasoning:
- The height when $t=0$ is $(\text-16(0)-2)(0-1) = (\text-2)(\text-1) = 2$ .
- (If students graph the function) The graph shows a $y$ -intercept of $(0,2)$ .

Lesson 15

Vertex Form

Visualizing A Graph

Function $f$ is given by $f(x) = (x+3)^2 -1$ .

Write the coordinates of the vertex of the graph of $f$ .
Does the graph open upward or downward? Explain how you know.

Show Solution

The vertex of $f$ is at $(\text-3,\text-1)$ .
The graph opens upward. The squared term has a positive coefficient (1).

Lesson 17

Changing the Vertex

Nudging a Graph

Here is a graph that represents $y = x^2$ . On the same coordinate plane, sketch a graph that represents $y = (x-6)^2 + 10$ .
The graph representing $y = x^2$ is shifted 2 units to the left, 10 units down, and flipped so that it opens downward, as shown. Write an equation that defines this curve.
Two parabolas in x y plane, origin O. X axis negative 6 to 6, by 2’s. Y axis negative 30 to 30, by 10s. First parabola opens upward with vertex at the origin. Second parabola opens downward with vertex at negative 2 comma negative 10.

Show Solution

A graph the same shape as $y = x^2$ but shifted 6 units to the right and 10 units up. Its vertex is at $(6,10)$ .
$y = \text-(x+2)^2-10$

Section D Check

Section D Checkpoint

Problem 1

For each pair of functions, describe how the graphs are different.

$f(x) = x^2 + 3$ and $g(x) = \text{-}x^2 +3$
$h(x) = x^2 + 6$ and $j(x) = x^2 + 1$
$m(x) = x^2 + 1$ and $p(x) = (x-2)^2 + 1$
$r(x) = x^2$ and $s(x) = 2x^2$

Show Solution

Sample responses:

The graph of $g$ has the same shape as the graph of $f$ , but is flipped to open downward.
The graph of $j$ has the same shape as the graph of $h$ , but is shifted 5 units down.
The graph of $p$ has the same shape as the graph of $m$ , but the graph of $p$ is 2 units to the right.
The graph of $s$ is taller and narrower than the graph of $r$ .

Problem 2

For each function, find the vertex, and then state whether it is a maximum or minimum.

$A(x) = \text{-}(x-3)^2 + 2$
$B(x) = 2(x+1)^2 + 3$
$C(x) = \text{-}5(x+2)^2$
$D(x) = x^2 - 8$

Show Solution

$(3,2)$ maximum
$(\text{-}1,3)$ minimum
$(\text{-}2,0)$ maximum
$(0,\text{-}8)$ minimum

Unit 7 Introduction To Quadratic Functions — Unit Plan