Task Sets: Explain intersection as solution - Common Core Math - HS Algebra

Work through problems with immediate feedback

Recall / Warm-Up

To solve $f(x) = g(x)$ graphically, what should you graph?

Only $y = f(x)$ and find its x-intercepts.

Both $y = f(x)$ and $y = g(x)$ on the same coordinate plane, then find intersections.

The single equation $y = f(x) \cdot g(x)$ and find its x-intercepts.

Only $y = g(x)$ and find its y-intercepts.

$Coordinate plane showing two curves intersecting at the point (3, 7). Dashed lines drop from the intersection to x = 3 on the x-axis and to y = 7 on the y-axis. The x-axis label shows 'solution = x = 3' and the y-axis label shows 'shared output = 7'.$

The graphs of $y = f(x)$ and $y = g(x)$ intersect at the point $(3, 7)$ .

What is the solution to the equation $f(x) = g(x)$ ?

$x = 3$

$x = 7$

$x = 3$ and $y = 7$

$(3, 7)$

A student graphs $y = 2x + 1$ and $y = 5$ on the same axes. The two graphs intersect at the point $(2, 5)$ .

What can the student conclude?

$x = 5$ satisfies $2x + 1 = 5$ .

$x = 2$ satisfies $2x + 1 = 5$ .

There is no solution because $2x + 1 \neq 5$ for any real $x$ .

$y = 5$ is the solution to $2x + 1 = 5$ .

Fluency Practice

The graphs of $y = x^2$ and $y = x + 2$ intersect at the points $(-1, 1)$ and $(2, 4)$ .

(a) State the solutions to $x^2 = x + 2$ .
(b) Explain in one sentence why the x-coordinates — not the y-coordinates — are the solutions.

Set up the graphical approach to solve $x^2 - 3x = -2$ .

(a) Identify $f(x)$ and $g(x)$ .
(b) Describe what you would graph and what you would look for.
(c) Verify that $x = 1$ and $x = 2$ are the solutions using substitution.

A student graphs $y = x^2$ and $y = 4$ on the same axes. How many solutions does $x^2 = 4$ have, and what are they?

One solution: $x = 2$

Two solutions: $x = 2$ and $x = -2$

No solutions, because a parabola and a horizontal line cannot intersect.

One solution: $x = 4$

Which of the following equations can be solved graphically using Desmos, even though it cannot easily be solved algebraically at this course level?

$2x + 1 = 7$

$x^2 - 5x + 6 = 0$

$2^x = x + 3$

$\frac{1}{2}x = 4$

A student uses Desmos to solve $|x - 1| = 2x - 3$ by graphing $y = |x - 1|$ and $y = 2x - 3$ . The graphs appear to intersect at $x = 2$ and the student finds no other intersection in the standard window.

What should the student do before accepting $x = 2$ as the only solution?

Accept $x = 2$ immediately — once one solution is found, the equation is solved.

Zoom out to check whether the graphs intersect anywhere else, then verify each candidate by substitution.

Also check whether $x = -2$ is a solution, since absolute value equations always have two solutions.

Switch to the algebraic method to confirm there are no other solutions.

You're viewing 2 of 6 sections.

Create a free account to continue the full exercise set and save your progress.

Create free account

0 of 8 answered

Answer all problems to submit.

Exercises: Graph Intersections as Solutions

Recall / Warm-Up

Fluency Practice