Graphing from the Vertex Form: IM Alg1.6.16
[url=https://im.kendallhunt.com/HS/teachers/1/6/16/preparation.html]“Graphing from the Vertex Form”[/url] from IM Algebra I © 2019 [url=https://www.illustrativemathematics.org/]Illustrative Mathematics[/url]. Licensed under the [url=https://creativecommons.org/licenses/by/4.0/]Creative Commons Attribution 4.0[/url] license.
Table of Contents
- Alg1.6.16 Lesson
- IM Alg1.6.16 Lesson: Graphing from the Vertex Form
- Alg1.6.16 Practice
- IM Alg1.6.16 Practice: Graphing from the Vertex Form
IM Alg1.6.16 Lesson: Graphing from the Vertex Form
Expressions in different forms can be used to define the same function. Here are three ways to define a function f.
[table][tr][td][math]f\left(x\right)=x^2-4x+3[/math][br][/td][td](standard form)[/td][/tr][tr][td][math]f\left(x\right)=\left(x-3\right)\left(x-1\right)[/math][/td][td](factored form)[/td][/tr][tr][td][math]f\left(x\right)=\left(x-2\right)^2-1[/math][/td][td](vertex form)[/td][/tr][/table][br][size=150]Which form would you use if you want to find the following features of the graph of [math]f[/math]? Be prepared to explain your reasoning.[/size][br][br]the [math]x[/math]-intercepts
the vertex
the [math]y[/math]-intercept
Here are two equations that define quadratic functions.
[math]p\left(x\right)=-\left(x-4\right)^2+10[/math][br][math]q\left(x\right)=\frac{1}{2}\left(x-4\right)^2+10[/math][br][br][size=150]The graph of [math]p[/math] passes through [math]\left(0,-6\right)[/math] and [math]\left(4,10\right)[/math], as shown on the coordinate plane.[/size][br][br]Find the coordinates of another point on the graph of [math]p[/math]. Explain or show your reasoning. Then, use the points to sketch and label the graph.
On the same coordinate plane, identify the vertex and two other points that are on the graph of [math]q[/math]. Explain or show your reasoning. Sketch and label the graph of [math]q[/math].[br]
Priya says, "Once I know the vertex is [math]\left(4,10\right)[/math], I can find out, without graphing, whether the vertex is the maximum or the minimum of function [math]p[/math]. I would just compare the coordinates of the vertex with the coordinates of a point on either side of it."[br][br]Complete the table and then explain how Priya might have reasoned about whether the vertex is the minimum or maximum.
[size=150]Write the equation for a quadratic function whose graph has the vertex at [math]\left(2,3\right)[/math] and contains the point [math]\left(0,-5\right)[/math].[/size][br]
Sketch a graph of your function.
Your teacher will give you a set of cards.
Each card contains an equation or a graph that represents a quadratic function. Take turns matching each equation to a graph that represents the same function.[br][list][*]For each pair of cards that you match, explain to your partner how you know they belong together.[/*][*]For each pair of cards that your partner matches, listen carefully to their explanation. If you disagree, discuss your thinking and work to reach an agreement.[/*][*]Once all the cards are matched, record the equation, the label and a sketch of the corresponding graph, and write a brief note or explanation about how you knew they were a match.[/*][/list]
IM Alg1.6.16 Practice: Graphing from the Vertex Form
[size=150]Which equation can be represented by a graph with a vertex at [math]\left(1,3\right)[/math]?[/size]
[size=150]Where is the vertex of the graph that represents [math]y=\left(x-2\right)^2-8[/math]?[/size][br]
[size=150][size=100]Where is the [math]y[/math]-intercept? Explain how you know.[br][/size][/size]
Identify one other point on the graph of the equation. Explain or show how you know.[br]
Sketch a graph that represents the equation.
[size=150]The function [math]v[/math] is defined by [math]v(x)=\frac{1}{2}(x+5)^2-7[/math].[br][br][/size]Without graphing, determine if the vertex of the graph representing [math]v[/math] shows the minimum or maximum value of the function. Explain how you know.
Here is a graph that represents y=x².
[size=150]Describe what would happen to the graph if the original equation was changed to:[br][/size][br][math]y=\frac{1}{2}x^2[/math] [br]
[math]y=x^2-8[/math]
Graph the equation [math]y=\frac{1}{2}x^2-8[/math] on the same coordinate plane as [math]y=x^2[/math].[br]
Clare throws a rock into the lake.
The graph shows the rock's height above the water, in feet, as a function of time in seconds.[br][img]data:image/png;base64,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[/img][br]Select [b]all[/b] the statements that describe this situation.[br]
Two objects are launched into the air.
[list][*]The height, in feet, of Object A is given by the equation [math]f(t)=4+32t-16t^2[/math].[br][/*][*]The height, in feet, of the Object B is given by the equation [math]g(t)=2.5+40t-16t^2[/math]. In both functions, [math]t[/math] is seconds after launch.[/*][/list]
Use technology to graph each function in the same graphing window.
What is the maximum height of each object?
Which object hits the ground first? Explain how you know.[br]
[size=150]Andre thinks the vertex of the graph of the equation [math]y=(x+2)^2-3[/math] is [math](2,-3)[/math]. Lin thinks the vertex is [math](-2,3)[/math].[/size][br] Do you agree with either of them?
[size=150]The expression [math]2000\cdot(1.015^{12})^5[/math] represents the balance, in dollars, in a savings account.[/size][br][br]What is the rate of interest paid on the account?[br]
How many years has the account been accruing interest?[br]
How much money was invested?[br]
How much money is in the account now?[br]
Write an equivalent expression to represent the balance in the savings account.[br]