Solving Equations Using Graphs

An equation of the form $f(x) = g(x)$ can be solved by identifying the points where the graph of $y = f(x)$ intersects the graph of $y = g(x)$. The solutions to the equation are the x-coordinates of these intersection points.

When an equation is set to zero, such as $f(x) = 0$, the solutions are the x-intercepts (also known as roots) of the graph $y = f(x)$. These are the locations where the curve crosses the horizontal axis.

For simultaneous equations involving two variables, the solution is the entire coordinate pair $(x, y)$ where the two lines or curves meet, satisfying both equations at once.

Solving $f(x) = k$

• To solve an equation where a function equals a constant, plot the curve $y = f(x)$ and the horizontal line $y = k$. The solutions are the x-coordinates where they cross.

Solving $f(x) = g(x)$

• Plot both functions on the same set of axes. Identify every point where the lines intersect and read the corresponding values from the x-axis.

Rearranging for Existing Graphs

• If you are given a graph of $y = f(x)$ but need to solve a different equation like $f(x) + h(x) = 0$, rearrange it to $f(x) = -h(x)$. You then plot the new line $y = -h(x)$ on the existing axes to find the solutions.

• Roots vs. Intersections: A 'root' specifically refers to where a single function equals zero (x-intercept), while an 'intersection' refers to where two functions equal each other.

• Single Variable vs. Simultaneous: When solving for $x$ in $f(x)=g(x)$, only the x-coordinate is required. When solving simultaneous equations in $x$ and $y$, both coordinates of the intersection point must be provided.

• Check the Scale: Always determine the value of each small grid square on the axes before reading off solutions. Misreading the scale is the most common source of lost marks.

• Draw Clear Lines: Use a sharp pencil and a ruler for straight lines. For curves, ensure they are smooth and pass through all plotted points accurately.

• Number of Solutions: Before finishing, look at the degree of the equation (e.g., quadratic, cubic) to ensure you haven't missed an intersection point that might be off-screen or at the edge of the grid.

• Verification: If time permits, substitute your estimated x-value back into the original equation. The two sides should be approximately equal if your reading was accurate.

• The Y-Coordinate Trap: Students often mistakenly give the y-coordinate of the intersection as the solution. Unless solving simultaneous equations, the answer is always the x-value.

• Incomplete Rearrangement: When modifying an equation to fit a given graph, students often forget to apply the change to both sides of the equation, leading to the wrong line being plotted.

• Missing Roots: On reciprocal or periodic graphs (like trig functions), solutions can repeat or exist in different quadrants. Always check the specified range (e.g., $0 \le x \le 360$) to ensure all valid solutions are found.