Desmos: Graphing, Tables, Intersections, and Zeros

17.1: Desmos: Graphing, Tables, Intersections, and Zeros. Original editorial SAT preparation with three worked examples, ten flashcards, and thirty practice questions. Educator-review draft.

Lesson 1 of 63 parts Practice Flashcards

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Graph and table setup

Objective: Enter a function and read an output from a specified input.

Prerequisite skills: function notation, equations, and interpreting coordinates. In Desmos graphing mode, type a complete rule such as y=2x+3 or define f(x)=2x+3. Add a table to compare selected inputs and outputs. Use the same variable meanings and units as the problem; parentheses preserve grouped operations. Common error: reading x and y in reversed order. Restrictions and interpretation: A graph is a check of the entered expressions; verify exact values and domain restrictions algebraically.

Formulas

table point=(input, output)

Worked examples

Try itA Desmos table for f(x) = 4x − 3 contains an input x = 23. What output should appear?Show answer

Substitute the table input: f(23)=4(23)−3=89. The table row is (23,89). Therefore the requested answer is 89.

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Graph and table setup

Objective: Enter a function and read an output from a specified input.

Formulas

table point=(input, output)

Worked examples

Try itA Desmos table for f(x) = 4x − 3 contains an input x = 23. What output should appear?Show answer

Substitute the table input: f(23)=4(23)−3=89. The table row is (23,89). Therefore the requested answer is 89.

Intersections

Objective: Use two entered equations and identify the correct intersection coordinate.

Graph each side of an equation as an output expression, such as y=L(x) and y=R(x), and inspect their intersection. Its x-coordinate solves L(x)=R(x). Adjust the viewing window, then verify with algebra because a rounded coordinate is not an exact proof. Common error: reporting the shared y-value when x is requested. Restrictions and interpretation: A graph is a check of the entered expressions; verify exact values and domain restrictions algebraically.

Formulas

$L(x)=R(x) at an intersection$

Worked examples

Try itThe graphs y = 5x + 5 and y = 2x + 80 meet. What is the intersection x-coordinate?Show answer

Set the outputs equal: 5x+5=2x+80. Thus 3x=75, giving x=25. The shared y-value is 130; it is not the requested coordinate. Therefore the requested answer is 25.

Zeros and visible windows

Objective: Find zeros and account for restrictions rather than relying on the window.

An x-axis crossing solves f(x)=0. A repeated quadratic root can touch the axis without crossing. A graph window may omit a real zero, so use factoring or another exact method to confirm all allowed roots. Common error: assuming no visible crossing proves no real root. Restrictions and interpretation: A graph is a check of the entered expressions; verify exact values and domain restrictions algebraically. Desmos and method choice: Practice in the College Board Desmos testing calculator or Bluebook test preview. Graphing and scientific modes serve different tasks. Enter expressions with explicit grouping, use tables for discrete inputs, and verify an intersection or zero by substitution. A viewport alone does not establish the full solution set.

Formulas

$f(x)=a(x−r)(x−s) ⇒ zeros r,s$

Worked examples

Try itA graph of f(x) = (x − 27)(x − 32) is initially viewed only for −2 ≤ x ≤ 2. What is the greater real zero of f?Show answer

Factor form already gives zeros x=27 and x=32, even though neither appears in the initial window. The greater is 32. Enlarge the window to see the corresponding intercepts. Therefore the requested answer is 32.

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