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A.6(C)·Algebra II·Quadratic FunctionsSupporting

Quadratics from Roots

Determine a quadratic function from its roots or its graph.

Students construct quadratic functions from roots, emphasizing repeated roots and using an additional point when the leading coefficient is unknown.

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.

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Roots determine factors

If r is a zero, then x−rx-r is a factor. With roots r and s, a quadratic has form f(x)=a(x−r)(x−s)f(x)=a(x-r)(x-s), where a≠0a\ne 0. Roots alone do not usually determine a.

A repeated root

When a quadratic has just one distinct real zero r, both factors are x−rx-r. Thus f(x)=a(x−r)2f(x)=a(x-r)^2. The vertex lies on the x-axis.

Recover coefficients

For a monic repeated-root quadratic, (x−r)2=x2−2rx+r2(x-r)^2=x^2-2rx+r^2. If a is unknown, substitute a nonzero-output point into the factored rule before expanding.

Worked example

Find b so f(x)=x2+bx+81f(x)=x^2+bx+81 has −9 as its only zero.

  1. A zero of −9-9 requires the factor x+9x+9.
  2. The only zero is repeated, so f(x)=(x+9)2f(x)=(x+9)^2.
  3. Expand: (x+9)2=x2+18x+81(x+9)^2=x^2+18x+81.
  4. Compare the coefficients of x.

b=18b=18

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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 Solving Quadratic Equations

UT IPSI · Kid2Kid / Texas Education Agency

Connect quadratic roots to factors and repeated solutions.

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TEKSForge practice · unscored

  1. 1. Write a monic quadratic with zeros −3 and 8.

    Show answer and reasoning

    f(x)=x2−5x−24f(x)=x^2-5x-24

    (x+3)(x−8)=x2−5x−24(x+3)(x-8)=x^2-5x-24.

  2. 2. A quadratic has only zero 4 and passes through (0, 32). Write its rule.

    Show answer and reasoning

    f(x)=2(x−4)2f(x)=2(x-4)^2

    Use f(x)=a(x−4)2f(x)=a(x-4)^2. The point gives 32=16a32=16a, so a=2a=2.

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 40 · 1 point

For what value of bb will f(x)=x2+bx+400f(x) = x^2 + bx + 400 have −20 as its only zero? Record your answer.

Show STAAR answer and explanation

40

A single zero at −20 means f(x)=(x+20)2=x2+40x+400f(x)=(x+20)^2=x^2+40x+400, so b=40b=40. Check: the discriminant is 402−4(400)=040^2-4(400)=0.

-40: b=−40b=-40 gives (x−20)2(x-20)^2, whose only zero is +20.

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Learning objectives

By the end of the lesson, students can:

  • Translate a zero into a linear factor.
  • Construct a quadratic with a repeated root.
  • Determine a leading coefficient from an additional point.

Instructional sequence

  1. 1

    Engage

    5 min

    Compare a parabola crossing the x-axis twice with one touching it once.

  2. 2

    Direct instruction

    10 min

    Connect each root to a factor and represent a repeated root with a squared factor.

  3. 3

    Guided practice

    12 min

    Build and expand monic quadratics with specified roots.

  4. 4

    Graph connection

    12 min

    Use another graph point to determine the leading coefficient.

  5. 5

    Closure

    6 min

    Students check the repeated-root coefficient in the exit ticket.

Misconceptions to address

““A root of −6 gives the factor x − 6.””

Teacher move: A root r gives x−rx-r. For r=−6r=-6, the factor is x+6x+6.

““One real zero makes the function linear.””

Teacher move: A quadratic can have the same root twice. Its graph touches the x-axis at the vertex.