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A.8(C)·Algebra II·Quadratic & Square Root FunctionsSupporting

Solutions from Quadratic Graphs

Use graphs to determine and interpret solution sets of quadratic equations.

Students connect the zeros of a quadratic graph to the solution set of f(x)=0f(x)=0, explain why a quadratic can have two, one, or no real solutions, and verify graphical answers algebraically.

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

Learn the concept

Zeros are solutions

To solve f(x)=0f(x)=0, find the inputs that make the output 0. On a graph, those points lie on the x-axis. If a parabola crosses the axis at (−2,0)(-2,0) and (4,0)(4,0), the solution set is {−2,4}\{-2,4\}.

Two, one, or none

A parabola can cross the x-axis twice, touch it once at the vertex, or never reach it. These cases correspond to two real solutions, one repeated real solution, and no real solutions.

Verify the graph

Graphs show approximate locations. Confirm values by substitution or by matching factors: zeros at −2 and 4 correspond to factors (x+2)(x+2) and (x−4)(x-4).

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 4 · 1 point

An upward-opening parabola crosses the x-axis at approximately 7.5 and 17.5 and has its vertex near 12.5, negative 25.
Quadratic function fOriginal visual from the released STAAR test.

Worked example

The graph of f(x)=x2−2x−8f(x)=x^2-2x-8 crosses the x-axis at two grid points. Find and verify the solution set of f(x)=0f(x)=0.

  1. The graph meets the x-axis at (−2, 0) and (4, 0).
  2. Factor the rule: x2−2x−8=(x−4)(x+2)x^2-2x-8=(x-4)(x+2).
  3. Each factor is zero at x=4x=4 or x=−2x=-2, matching the graph.

{−2,4}\{-2,4\}

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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

Read and interpret key points on quadratic graphs, including where the graph meets the x-axis.

Watch on YouTube
Texas public school video

Algebra 2 Solving Quadratic Equations

UT IPSI · Kid2Kid / Texas Education Agency

Solve quadratic equations and connect the algebraic solutions to their graphs.

Watch on YouTube

Try it yourself

TEKSForge practice · unscored

  1. 1. A parabola opens upward and has vertex (3, 0). How many real solutions does f(x) = 0 have?

    Show answer and reasoning

    One real solution, x = 3

    The vertex lies on the x-axis and is the lowest point, so the graph touches the axis only once.

  2. 2. A parabola opens upward and has vertex (1, 5). How many real solutions does f(x) = 0 have?

    Show answer and reasoning

    No real solutions

    The lowest output is 5, so the graph never reaches y = 0.

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 4 · 1 point

An upward-opening parabola crosses the x-axis at approximately 7.5 and 17.5 and has its vertex near 12.5, negative 25.
Quadratic function fOriginal visual from the released STAAR test.

The graph of the quadratic function ff is shown on the grid. What does the solution set for f(x)=0f(x) = 0 appear to be?

  1. {12.5}\{12.5\}
  2. {12.5,−25}\{12.5, -25\}
  3. {7.5,17.5}\{7.5, 17.5\}
  4. {7.5,12.5,17.5}\{7.5, 12.5, 17.5\}
Show STAAR answer and explanation

{7.5,17.5}\{7.5, 17.5\}

The solutions to f(x)=0f(x)=0 are the x-coordinates where the graph crosses the x-axis. The parabola crosses at approximately 7.5 and 17.5.

{12.5}\{12.5\}: 12.5 is the x-coordinate of the vertex, where f(x)f(x) is about −25 rather than 0.

{12.5,−25}\{12.5, -25\}: These are the vertex coordinates, not the two x-intercepts.

{7.5,12.5,17.5}\{7.5, 12.5, 17.5\}: The graph has only two x-intercepts; 12.5 locates the axis of symmetry, not another zero.

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Start scored STAAR practice

Learning objectives

By the end of the lesson, students can:

  • Identify the solutions of f(x)=0f(x)=0 as the x-intercepts of the graph.
  • Explain how the vertex position determines the number of real solutions.
  • Verify graphical solutions by substitution or factoring.

Instructional sequence

  1. 1

    Engage

    5 min

    Show three parabolas that cross, touch, and miss the x-axis. Ask what each says about the equation f(x) = 0.

  2. 2

    Direct instruction

    10 min

    Connect y = 0 to the x-axis. Model reading intercepts precisely from grid lines.

  3. 3

    Guided practice

    12 min

    Read zeros from graphs, write the solution sets, and verify each solution by substitution.

  4. 4

    Compare methods

    12 min

    Match graphs to factored forms and explain why the graph and algebra give the same values.

  5. 5

    Closure

    6 min

    Students complete the exit ticket and name one graph with no real solutions.

Misconceptions to address

““The y-intercept is the solution to f(x) = 0.””

Teacher move: The y-intercept is where x = 0. Solutions to f(x) = 0 are the x-values where the output y is 0.

““Every quadratic has exactly two solutions.””

Teacher move: Show a parabola touching the x-axis at its vertex (one real solution) and one that does not reach the axis (no real solutions).