options: Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).

Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.

Graph the solutions to a linear inequality in two variables as a half plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.
problem_activity: Optional activity
This activity is optional because it goes beyond the depth of understanding required by the standards.
The purpose of this activity is to connect graphs of quadratic functions with solutions to related quadratic equations to tell how many solutions an equation has, and whether those solutions are real or non-real. Students have already solved two of the equations in the previous activity, and the equation they haven’t solved involves the perfect square that connects all three of the equations. Graphically,
\(y=f(x)\)
intersects the
\(x\)
-axis twice, which corresponds to the two real solutions to
\(x^2 - 8x + 13 = 0\)
. The graph
\(y=g(x)\)
intersects the
\(x\)
-axis once, which corresponds to the single real solution to
\(x^2 - 8x + 16 = 0\)
. However, the graph
\(y=h(x)\)
doesn’t intersect the
\(x\)
-axis, which reflects the fact that the complex solutions to
\(x^2 - 8x + 19 = 0\)
can’t be described with a real number line.
Launch
Provide access to graphing technology.
Student Facing
How many real solutions does each equation have? How many non-real solutions?
\(x^2 - 8x + 13 = 0\)
\(x^2 - 8x + 16 = 0\)
\(x^2 - 8x + 19 = 0\)
How do the graphs of these functions help us answer the previous question?
\(f(x) = x^2 - 8x + 13\)
\(g(x) = x^2 - 8x + 16\)
\(h(x) = x^2 - 8x + 19\)
Student Response
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Anticipated Misconceptions
Some students may attempt to plot the complex solutions in the coordinate plane as if it were the complex plane. Remind these students that points in the coordinate plane are pairs of real numbers
\((x,y)\)
, whereas points in the complex plane are single complex numbers. In other words, the point
\((2,4)\)
in the coordinate plane is a different kind of object than the point
\(2+4i\)
in the complex plane.
Activity Synthesis
Select students to share their solutions and discuss the connections between the graphs and the equations. Discuss the idea that if a graph of a quadratic function doesn’t cross a given horizontal line (in this case,
\(x=0\)
) this doesn’t indicate that the related equation has no solutions, but rather that it has no
real
solutions that can be graphed with real number lines.
Speaking: MLR8 Discussion Supports.
To help students produce statements about the connection between graphs of quadratic functions and solutions to related quadratic equations, provide sentence frames such as: “If _____ then _____ because…” and  “On the graph, I see _____, so I know….” Show concepts multi-modally by displaying the graphs of
\(f(x)\)
,
\(g(x)\)
, and
\(h(x)\)
.
Design Principle(s): Support sense-making


correct_option_index: 1