10. Reading Graphs and Intersections

On a graph, a system's solutions are the points where the graphs share the same (x)- and (y)-coordinates. SAT® questions may ask for the number of intersections, an intersection coordinate, or a value built from the coordinates.

If (f(x)=x^2-4) and (g(x)=x+2), intersections occur where
\[ f(x)=g(x). \]
The roots (x=-2) and (x=3) are the horizontal coordinates of the intersections. The corresponding vertical coordinates come from either function.

Graphical clues:

• Two crossings correspond to two distinct real roots.
• One touch point corresponds to a repeated root and discriminant zero.
• No crossing corresponds to a negative discriminant for the resulting quadratic.

Use the graph to estimate and the algebra to establish exact values. A graphing window can hide an intersection, so adjust the window or confirm algebraically.

Professor's tip: Read exactly what the question requests: number of solutions, an (x)-coordinate, a (y)-coordinate, an ordered pair, or an expression such as (x_1+x_2).

Common trap: Reporting the (x)-intercepts of one graph instead of the intersections of both graphs.

Session Goals

You will extract exact information from graphs, detect misleading windows and scales, and connect intersection behavior to signs of (f(x)-g(x)).

Think in Terms of a Difference Function

Define (h(x)=f(x)-g(x)). Intersections of (f) and (g) are precisely the zeros of (h). This viewpoint connects systems to everything you know about quadratic roots.

If (h(x)=2(x-1)(x-5)), then the graphs intersect at (x=1) and (x=5). Moreover:

• (h(x)>0) for (x<1) and (x>5), so (f(x)>g(x)) there.
• (h(x)<0) for (1<x<5), so (f(x)<g(x)) there.

Thus a factored difference tells not only where the graphs meet but also which graph lies above the other.

Graph Reliability

An apparent touch point may be two very close intersections. A window may exclude a distant root. Decimal intersection coordinates may conceal exact radicals or fractions. Use zooming and tables to form a conjecture, then use algebra when an exact answer or proof is required.

Advanced Checkpoint

Suppose (f(x)-g(x)=x^2-6x+k). A graph shows one intersection. Then (D=36-4k=0), so (k=9), and the intersection occurs at (x=3). This combines graphical evidence with exact algebra.

Ten Worked Examples

Example 1

Question: If (f-g=(x-2)(x-5)), where do the graphs intersect?

Solution: Intersections occur where (f-g=0), so (x=2) and (x=5).

Example 2

Question: For the same difference, where is (f<g)?

Solution: The upward-opening product is negative between its roots. Therefore (f<g) for (2<x<5).

Example 3

Question: If (f-g=-(x+1)(x-4)), where is (f>g)?

Solution: The expression is positive between roots (-1) and (4). Thus (f>g) for (-1<x<4).

Example 4

Question: A graph touches once and the difference is (x^2-6x+k). Find (k).

Solution: One intersection means (D=0): (36-4k=0), so (k=9).

Example 5

Question: Where does that touch occur?

Solution: With (k=9), the difference is ((x-3)^2). The touch occurs at (x=3).

Example 6

Question: If a graph window shows no intersection, is that proof?

Solution: No. An intersection may lie outside the window. Expand the window, inspect a table, or solve algebraically.

Example 7

Question: The graph reports (x\approx2.236). What exact value is suggested?

Solution: Since (2.236\approx\sqrt5), test whether the algebra produces (x^2=5). A decimal graph reading is evidence, not proof.

Example 8

Question: If (h(x)=f(x)-g(x)) has a double root, what happens graphically?

Solution: The sign of (h) does not change at the root, so the graphs touch but do not cross there.

Example 9

Question: If (h(x)) changes from positive to negative at (x=4), what occurs?

Solution: At (x=4), (h=0) and the relative order reverses, so the graphs cross at that (x)-coordinate.

Example 10

Question: A graph shows intersections at (x=-2,3). Build a possible monic difference function.

Solution: A monic quadratic with those roots is (h(x)=(x+2)(x-3)=x^2-x-6).