A.6(B)·Algebra II·Quadratic FunctionsReadiness

Representations of Quadratics

Relate algebraic, tabular, graphical, and verbal representations of quadratic functions.

Students connect quadratic equations, tables, graphs, and verbal descriptions by identifying the vertex, symmetry, intercepts, and changing rate of growth.

Lesson plan and worked examples by TEKSForge · not a TEA-issued lesson. Videos by independent educators.

Archived course: TEKS code and readiness label from TEA's 2013 Algebra II key. Algebra II is not one of Texas's current required STAAR EOC assessments, and this lesson does not change the Biology and Algebra I Study Plans.

Practice the released item Ask the tutor All Algebra II released practice50 minutes

Learn the concept

Build a rule from the vertex

A vertex $(h,k)$ gives $y=a(x-h)^2+k$ . Substitute another point to determine a. The vertex alone does not determine the width or direction of opening.

Read symmetry in a table

Distinct inputs with equal outputs lie equally far from the axis. If $g(1)=g(9)$ , the axis is $x=5$ . Reflect points with $x_{\text{new}}=2h-x_{\text{old}}$ , preserving the output.

Connect shape to context

An area model such as $A(s)=3s^2$ has vertex at the origin and opens upward. In the physical domain $s\ge 0$ , it rises more steeply as s increases. Tables at equally spaced inputs have constant second differences.

Study released STAAR visuals

Use the original graphs, tables, and diagrams from released STAAR questions to practice reading labels, values, and relationships before answering the items below.

Archived released STAAR Algebra II EOC · 2013 · Item 33 · 1 point

A quarter-circle sector with a right angle at its corner. Both straight sides are labeled r, the radius. — Quarter-circle sector with radius rOriginal visual from the released STAAR test.

Worked example

A parabola has vertex (−2, 3) and passes through (0, 11). Write its rule and describe a table pattern that confirms its symmetry.

Use the vertex: $y=a(x+2)^2+3$ .
Substitute $(0,11)$ : $11=4a+3$ , so $a=2$ .
The rule is $y=2(x+2)^2+3$ .
Inputs equally far from $-2$ match: $f(-3)=f(-1)=5$ and $f(-4)=f(0)=11$ .

$f(x)=2(x+2)^2+3$ , with axis $x=-2$ and matching outputs at reflected inputs.

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Texas public-school, Texas educator-created, and TEKS-aligned lessons, hosted by their creators. Each card names its source.

Texas public school video

Algebra 2 Interpreting Quadratic Graphs

UT IPSI · Kid2Kid / Texas Education Agency

Connect vertices, symmetry, intercepts, and quadratic representations.

Watch on YouTube

Try it yourself

TEKSForge practice · unscored

1. A quadratic table includes $(0,-7)$ , $(2,5)$ , $(4,9)$ , $(6,5)$ , and $(8,-7)$ . Identify its vertex and y-intercept.
Show answer and reasoning
Vertex $(4,9)$ ; y-intercept $(0,-7)$ .
Matching outputs show symmetry about x = 4. The table includes the vertex on that axis and the point whose input is 0.
2. Describe the graph of $A(r)=\frac{\pi}{3}r^2$ for $r\ge 0$ .
Show answer and reasoning
It begins at the origin and rises more steeply as r increases.
The positive coefficient gives an upward-opening parabola. The physical restriction keeps only the nonnegative-input portion.

Released STAAR practice

Texas Education Agency · archived 2013 released EOC items · scored with TEA's answer key; step-by-step explanations written by TEKSForge because this release has no rationales. Original graphs, tables, and diagrams are shown where the released item includes one.

Archived released STAAR Algebra II EOC · 2013 · Item 12 · 1 point

Which function's graph has a vertex at $(3, 5)$ and contains the point $(5, 13)$ ?

$y = \frac{1}{10}(x + 3)^2 - 5$
$y = \frac{1}{10}(x - 3)^2 - 5$
$y = 2(x - 3)^2 + 5$
$y = 2(x + 3)^2 + 5$

Show STAAR answer and explanation

$y = 2(x - 3)^2 + 5$

A vertex of $(3,5)$ gives $y=a(x-3)^2+5$ . Substitute $(5,13)$ : $13=a(5-3)^2+5$ , so $4a=8$ and $a=2$ .

$y = \frac{1}{10}(x + 3)^2 - 5$ : This has vertex $(-3,-5)$ ; both coordinates of the vertex have the wrong sign.

$y = \frac{1}{10}(x - 3)^2 - 5$ : This has vertex $(3,-5)$ , and at $x=5$ it gives $0.4-5=-4.6$ , not 13.

$y = 2(x + 3)^2 + 5$ : This has vertex $(-3,5)$ ; $(x+3)$ shifts the graph left, not right.

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Archived released STAAR Algebra II EOC · 2013 · Item 22 · 1 point

A table of values for the quadratic function $g$ is shown. Which of the following statements about the graph of $g$ are true? I. The graph has a line of symmetry at $x = 7$ . II. The graph has a $y$ -intercept at $(0, -40)$ . III. The graph has an $x$ -intercept at $(4, 0)$ . IV. The graph has a vertex at $(6, 8)$ .

Values of g
x	g(x)
−1	−55
2	−16
5	5
6	8
9	5
10	0
12	−16
14	−40

I and II only
III and IV only
I, II, and III only
I, II, III, and IV

Show STAAR answer and explanation

I, II, and III only

Equal outputs pair up around $x=7$ : $g(5)=g(9)=5$ and $g(2)=g(12)=-16$ , so I is true. Reflecting $(14,-40)$ across $x=7$ gives $(0,-40)$ , so II is true. Reflecting the zero at $x=10$ gives a zero at $x=4$ , so III is true. The vertex must lie on $x=7$ , so IV is false.

I and II only: Statement III is also true: the zero at $x=10$ reflects across $x=7$ to $x=4$ .

III and IV only: Statement IV is false because the vertex lies on the axis $x=7$ ; $(6,8)$ is just a point in the table. Statements I and II are true.

I, II, III, and IV: The vertex must lie on the axis of symmetry, $x=7$ , so $(6,8)$ cannot be the vertex.

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Archived released STAAR Algebra II EOC · 2013 · Item 33 · 1 point

A sector of a circle is shown. The area, $A$ , of the sector is $\frac{\pi}{4}$ times the square of the radius, $r$ . Which graph represents this relationship?

Graph A
Graph B
Graph C
Graph D

Show STAAR answer and explanation

Graph B

$A=\frac{\pi}{4}r^2$ is a quadratic function of r with a positive coefficient. It starts at the origin, increases, and rises more steeply as r increases—the upward-curving Graph B.

Graph A: Graph A decreases to zero as the radius grows, but a larger radius gives a larger area.

Graph C: Graph C starts above zero and decreases; a sector with radius 0 has area 0.

Graph D: Graph D increases but levels off, like a square root. $r^2$ grows faster and faster.

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Start scored STAAR practice

Learning objectives

By the end of the lesson, students can:

Write vertex form from a vertex and another point.
Use equal table outputs to locate an axis of symmetry.
Match a quadratic model to its graph and contextual behavior.

Instructional sequence

1
Engage
5 min
Compare a graph that grows more steeply with one that rises but levels off.
2
Direct instruction
12 min
Model finding a from a known vertex and another point.
3
Table investigation
13 min
Find matching outputs, calculate midpoint inputs, and infer reflected intercepts.
4
Representation sort
13 min
Match area contexts, tables, equations, and graphs using at least two features.
5
Closure
7 min
Students explain one match and complete the exit ticket.

Misconceptions to address

““The greatest listed output must be the vertex.””

Teacher move: A table may omit the vertex. Locate the symmetry axis before identifying the vertex.

““Every increasing curve represents a positive quadratic.””

Teacher move: A positive quadratic becomes steeper to the right of its vertex; a square-root curve typically levels off.

Materials

•Quadratic representation cards
•Coordinate grids
•Graphing calculator

Teaching tips

Require the vertex and one additional point when checking a proposed equation.
Draw arrows between equal outputs in a table.
Use the origin and curvature to distinguish positive area models from square-root graphs.

Exit ticket · 3–5 min

A quadratic has $g(1)=g(9)$ and a zero at $x=2$ . Find its axis and other zero.

Show expected answer

The axis is $x=\frac{1+9}{2}=5$ . Reflecting 2 across 5 gives the other zero, $x=8$ .

Sources & alignment

TEA 2013 Algebra II released test TEA 2013 Algebra II answer key and standards Texas Gateway, home of the UT IPSI Kid2Kid lessons

Alignment follows the standard TEA assigned to each 2013 released item. Video playback depends on YouTube access; written instruction remains available here.

Learn the concept

Build a rule from the vertex

Read symmetry in a table

Connect shape to context

Study released STAAR visuals

Archived released STAAR Algebra II EOC · 2013 · Item 33 · 1 point

Worked example

Watch the lesson

Algebra 2 Interpreting Quadratic Graphs

Try it yourself

1. A quadratic table includes (0,−7)(0,-7)(0,−7), (2,5)(2,5)(2,5), (4,9)(4,9)(4,9), (6,5)(6,5)(6,5), and (8,−7)(8,-7)(8,−7). Identify its vertex and y-intercept.

2. Describe the graph of A(r)=π3r2A(r)=\frac{\pi}{3}r^2A(r)=3π​r2 for r≥0r\ge 0r≥0.

Released STAAR practice

Archived released STAAR Algebra II EOC · 2013 · Item 12 · 1 point

Archived released STAAR Algebra II EOC · 2013 · Item 22 · 1 point

Archived released STAAR Algebra II EOC · 2013 · Item 33 · 1 point

Learning objectives

Instructional sequence

Engage

Direct instruction

Table investigation

Representation sort

Closure

Misconceptions to address

1. A quadratic table includes $(0,-7)$ , $(2,5)$ , $(4,9)$ , $(6,5)$ , and $(8,-7)$ . Identify its vertex and y-intercept.

2. Describe the graph of $A(r)=\frac{\pi}{3}r^2$ for $r\ge 0$ .