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A.9(A)·Algebra II·Square Root FunctionsSupporting

Square Root Transformations

Predict the effects of parameter changes on graphs of square root functions.

Students predict square-root graph transformations by separating input changes from output changes and tracking the endpoint.

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 tutorAll Algebra II released practice45 minutes

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The endpoint anchors the graph

The parent y=xy=\sqrt{x} begins at (0,0)(0,0). In y=ax−h+ky=a\sqrt{x-h}+k, the endpoint is (h,k)(h,k) and the domain is x≥hx\ge h.

Inside changes affect inputs

Replacing x with x−hx-h shifts right h units when h is positive. Replacing x with −x-x reflects the parent across the y-axis, producing a graph whose domain is x≤0x\le 0.

Outside changes affect outputs

A negative a reflects across the x-axis. Its magnitude stretches or compresses vertically. Adding k outside shifts the graph vertically. A parent point (u,u)(u,\sqrt u) becomes (u+h,au+k)(u+h,a\sqrt u+k).

Worked example

Describe and check the transformations from f(x)=xf(x)=\sqrt{x} to g(x)=−2x−6+1g(x)=-2\sqrt{x-6}+1.

  1. The expression x−6x-6 shifts the graph right 6 units.
  2. Multiplication by −2 reflects across the x-axis and stretches vertically by a factor of 2.
  3. Adding 1 shifts the result up 1, placing the endpoint at (6, 1).
  4. The parent point (4,2)(4,2) becomes (10,−3)(10,-3); indeed, g(10)=−24+1=−3g(10)=-2\sqrt4+1=-3.

Right 6, reflect across the x-axis, stretch vertically by 2, and move up 1.

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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 Connecting Quadratic and Square Foot Functions

UT IPSI · Kid2Kid / Texas Education Agency

Connect square-root graph features and their relationship to quadratic graphs.

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TEKSForge practice · unscored

  1. 1. Describe h(x)=−x+4h(x)=-\sqrt{x+4}.

    Show answer and reasoning

    Shift left 4 and reflect across the x-axis.

    The endpoint is (−4, 0). The outside negative reverses output signs.

  2. 2. Compare the domains of p(x)=−xp(x)=\sqrt{-x} and q(x)=−xq(x)=-\sqrt{x}.

    Show answer and reasoning

    p has domain x≤0x\le 0; q has domain x≥0x\ge 0.

    The radicand determines the domain. An outside negative does not change allowed inputs.

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 17 · 1 point

Which two transformations can be used to obtain the graph of g(x)=−x−cg(x) = -\sqrt{x - c} from the graph of f(x)=xf(x) = \sqrt{x} if c>0c > 0?

  1. A translation to the right cc units followed by a reflection across the xx-axis
  2. A translation to the left cc units followed by a reflection across the xx-axis
  3. A translation to the right cc units followed by a reflection across the yy-axis
  4. A translation to the left cc units followed by a reflection across the yy-axis
Show STAAR answer and explanation

A translation to the right cc units followed by a reflection across the xx-axis

Replacing x with x−cx-c shifts the graph right c units. The negative sign outside the radical multiplies every output by −1, reflecting the graph across the x-axis.

A translation to the left cc units followed by a reflection across the xx-axis: A left shift comes from x+c\sqrt{x+c}. Subtracting c inside the radical moves the graph right.

A translation to the right cc units followed by a reflection across the yy-axis: A reflection across the y-axis would require −x\sqrt{-x}. The negative sign outside the radical reflects across the x-axis.

A translation to the left cc units followed by a reflection across the yy-axis: Both the direction of the shift and the axis of reflection are reversed.

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Learning objectives

By the end of the lesson, students can:

  • Identify horizontal and vertical translations of a square-root graph.
  • Distinguish reflections across the x-axis and y-axis.
  • Verify transformations with an endpoint and another point.

Instructional sequence

  1. 1

    Engage

    5 min

    Plot three perfect-square inputs on the square-root parent graph.

  2. 2

    Direct instruction

    10 min

    Separate changes inside the radical from multipliers and shifts outside it.

  3. 3

    Guided practice

    12 min

    Track endpoints and points through translations and reflections.

  4. 4

    Compare rules

    12 min

    Contrast an outside negative sign with a negative input inside the radical.

  5. 5

    Closure

    6 min

    Students justify the endpoint and reflection in the exit ticket.

Misconceptions to address

““Subtracting inside the radical moves the graph left.””

Teacher move: Set the radicand equal to zero. The new endpoint shows the horizontal direction.

““A negative sign outside reflects across the y-axis.””

Teacher move: An outside negative changes outputs, so it reflects across the x-axis.