Combination of Graphs (Junior Cert Mathematics): Revision Notes

Step-by-Step Example: Question: Find the points where the functions $f(x) = x^2 - 2x$ and $g(x) = x - 2$ intersect.

1. Draw the Graphs:

• Start by sketching the graph of $f(x) = x^2 - 2x$. This is a U-shaped curve (called a parabola) that opens upwards.
• Next, draw the graph of $g(x) = x - 2$. This is a straight line.

2. Identify the Intersection Points:

• Look for the points where the two graphs cross each other. These are the points where both functions have the same value.

3. Read the Intersection Points from the Graph:

• From the graph, you'll see that the graphs intersect at two points: $x = 1$ and $x = 2$. Answer: The points of intersection are $x = 1$ and $x = 2$.

Why This Works: At these points, both functions give the same output (y$$-value) for the same input (x$$-value), which means they intersect on the graph.

Step-by-Step Example: Question: Find where $f(x) = x^2 - 6x + 5$ is less than $g(x) = x - 1$.

4. Draw the Graphs:

• Start by sketching the graph of$f(x) = x^2 - 6x + 5$. This is another U-shaped curve (parabola) that opens upwards.
• Then, draw the graph of $g(x) = x - 1$. This is a straight line.

5. Look for the Overlaps:

• Find the sections where the graph of $f(x)$ is below the graph of $g(x)$.

6. Identify the x-values:

• The graph of $f(x)$ is below the graph of $g(x)$ between the points of intersection, which occur at $x = 1$ and $x = 6$. Answer: The function $f(x)$ is less than $g(x)$for $1 < x < 6$.

Why This Works: Where the graph of $f(x)$ lies below $g(x)$, it means that for those $x-values$, the outputs of $f(x)$ are smaller than the outputs of $g(x)$.

This version uses the functions $f(x) = x^2 - 6x + 5$ and $g(x) = x - 1$, which intersect at $x = 1$ and $x = 6$, providing non-decimal intersection points.