Unit 2, Lesson 11
Finding Intersections
Integrated Math 3
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When asked to find all values of that make an equation like true, one way to consider the question is to ask where the graphs of the functions and intersect.
Functions f of x and g of x on an x y coordinate plane, no grid. Horizontal axis from negative 6 to 9, by 1's. Vertical axis from negative 50 to 10, by 10s. Starting in quadrant 2, f of x moves downwards, passing through negative 4 comma 0, intersects g of x at negative 1 comma negative 27, curves and moves upwards, intersecting g of x, and passing through 8 comma 0. Starting in quadrant 3, g of x moves upwards and to the right, intersects f of x at negative 1 comma negative 27, curves and moves downwards, intersecting f of x and 8 comma 0.
Since the coordinate of any point of intersection has the form , these points must make true when . In our example, we can tell from the graph that both and are solutions to the original equation.
We can also use algebra to identify solutions to by rearranging and then recognizing that both parts have a factor of in common:
For polynomials created to model specific situations that have a more complicated structure, solving without using technology can be challenging, especially because the graphs of two polynomials can intersect at multiple points.