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Algebra 2 9-12 / Week 08 / Thursday
4/6
Week 08 · The Function Toolkit

Thursday

Where two graphs meet
// Build, move, compare, cross
⏱ about 20 min

Thursday: Where Two Graphs Meet

Wren draws y = x² and the line y = 2x + 3 on the wall. They cross twice.

"The crossings are at x = -1 and x = 3," he says, pointing. "What do you notice about the two rules there?"

"They give the same y," Comet says. "At x = 3, both are 9."

"So a crossing is a solution of x² = 2x + 3," Wren says. "The picture and the equation agree."

Nova projects a table beside the graph. "Would you like a hint? When the graph is fuzzy, a table can close in."

"Close in?" Comet asks. "Try x = 2, then 2.5, then 2.6, and watch which side is bigger," Wren says.

"Three ways to the same point," Comet says. "Graph, table, algebra. Today we use all three."

Why a crossing is a solution

Two graphs are y = f(x) and y = g(x). At a crossing, the point is on both graphs.

So its y equals f(x) and also equals g(x). That x makes f(x) = g(x) true: it is a solution.

  1. A point (a, b) lies on the graph of y = f(x) exactly when b = f(a).
  2. The same point lies on the graph of y = g(x) exactly when b = g(a).
  3. A crossing point lies on both graphs, so b = f(a) and b = g(a).
  4. Two things equal to b are equal to each other: f(a) = g(a).
  5. So a, the x-coordinate of the crossing, is a solution of f(x) = g(x).
  6. The steps reverse too: any solution a gives a point (a, f(a)) on both graphs.
The parabola y = x squared and the line y = 2x + 3 cross at two marked points.

Way one: algebra

  1. Set the rules equal: x² = 2x + 3.
  2. Move everything to one side: x² - 2x + 3 = 0, which is x² - 2x - 3 = 0.
  3. Factor: (x - 3)(x + 1) = 0.
  4. So x = -1 or x = 3. Put each back into either rule for its y.

Way two: a table that closes in

When the crossing is not a whole number, a table brackets it. Here is x² = x + 4 between x = 2 and x = 3.

xx²x + 4difference
246-2
2.56.256.5-0.25
2.66.766.60.16
3972

The difference changes sign between 2.5 and 2.6, so a crossing lies there. Narrow the steps to get closer.

This is how a calculator finds a crossing it cannot factor: smaller and smaller steps until the difference is tiny.

A line meets a parabola

Any line and parabola meet twice, once or never. Set the rules equal and read the discriminant from week 1.

Take y = x² - 4x + 7 and y = x + 3. Setting the rules equal gives x² - 5x + 4 = 0.

That factors as (x - 1)(x - 4) = 0, so x = 1 or x = 4.

THE PROOF, IN ORDER
  • ?A point (a, b) is on y = f(x) exactly when b = f(a)
  • ?The same point is on y = g(x) exactly when b = g(a)
  • ?So the x-coordinate a solves f(x) = g(x)
  • ?A crossing is on both graphs, so b = f(a) and b = g(a)
  • ?Two things equal to b are equal: f(a) = g(a)
WHY THIS EXERCISEThis chain is why a graph can solve an equation. Every link is the definition of a graph.
A point where two graphs meet is called this. Type one word.
The x-coordinate of a crossing is a this of f(x) = g(x). Type one word.
b² - 4ac tells how many times a line meets a parabola. Its name is the this. Type one word.
FIND THE CROSSINGS
  • Read the question.
  • Tap your answer.
Where do y = x² and y = 2x + 3 meet?
Where do y = x² - 4x + 7 and y = x + 3 meet?
Where do y = x² - 2x and y = x + 4 meet?
In the table, x² - (x + 4) changes sign where?
StatementTrue or false?
At a crossing, f(x) and g(x) have the same value.?
3 × 3 = 2 × 3 + 3?
2 × 2 = 2 × 2 + 3?
A line and a parabola always meet exactly twice.?
A table of values can bracket a crossing between two x-values.?
WHY THIS EXERCISEGraph, table and algebra are three views of one fact. Checking one against another catches mistakes.

Clear reasoning. Tomorrow two functions given in different ways are compared head to head.

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