International Mathematics (0607) | Sketching graphs on a calculator

Welcome to Functions: Mastering the Graphic Display Calculator (GDC)

Hello! This chapter is all about using your most powerful mathematical tool—the Graphic Display Calculator (GDC)—to understand functions quickly and visually.

In IGCSE International Mathematics (0607), we study different types of functions, like linear (\(y = ax + b\)) and quadratic (\(y = ax^2 + bx + c\)). While we learn algebraic techniques to solve them, the GDC is essential for sketching complex or unfamiliar functions, and for locating key points accurately.

Think of your GDC as a mathematical GPS. It can draw the map (the sketch) and instantly tell you where the key landmarks are (zeros, turning points, intersections). Mastering these GDC skills will save you valuable time in your exams!

Part 1: The Essentials – Sketching and Producing Values

The first job of your GDC is simple: take the equation you provide and show you what it looks like.

1. Sketching the Graph of a Function

The syllabus requires you to use the GDC to sketch the graph of a function.

A sketch is not meant to be perfectly accurate or to scale. Instead, it must clearly show the most important features of the function.

Key Steps for Sketching (Using the GDC):

1. Enter the function (e.g., \(f(x) = x^3 - 4x\)) into the graphing editor (\(Y=\) screen).
2. Adjust the Window settings (X-min, X-max, Y-min, Y-max) so that the interesting parts (like turning points and intercepts) are visible.
3. Press Graph.
4. When drawing your sketch on paper, you must:
   
   • Draw the curve freehand and smoothly.
   • Label the axes (\(x\) and \(y\)).
   • Mark the coordinates or values of the key features (like intercepts or turning points, which you will find using the analysis tools described in Part 2).

Accessibility Tip: Window Settings
If your graph looks like a straight line or nothing at all, your window is probably too small or too large! Use the "Zoom Fit" or "Auto Zoom" function if your calculator has one, or try a standard view like \(X\) and \(Y\) limits from \(-10\) to \(10\).

2. Producing a Table of Values and Plotting Points

While the sketch gives you the overall shape, sometimes you need specific, accurate coordinates to plot points precisely on grid paper.

Steps to Get a Table of Values:

1. Enter the function (\(Y=\)).
2. Go to the Table function (often labeled as TABLE or TBLSET).
3. Set your Table Start value (e.g., -5) and the step size (\(\Delta Tbl\), usually 1).
4. The calculator will display a list of coordinates \((x, y)\).

Plotting Points (Syllabus Requirement):
If a question asks you to plot a graph, you must use the coordinates from your table (or the GDC's trace function) and mark them clearly on the graph paper, often using small crosses (\(x\)) or dots, ensuring accuracy within half a square on the grid.

Part 2: The Power Features – Analyzing Functions

The real power of the GDC is its ability to perform instantaneous analysis. It can calculate important points that would take ages (or require complex differentiation) to find algebraically. These functions are usually found under the CALC or G-SOLVE menu.

3. Finding Zeros (Roots/x-intercepts)

The zeros of a function are the \(x\)-values where the graph crosses the \(x\)-axis. At these points, the \(y\) coordinate is zero, so \(f(x) = 0\).

Analogy: If the graph is a roller coaster, the zeros are the points where the track meets the ground level.

GDC Steps (Generic):

1. Graph the function.
2. Select the G-SOLVE or CALC menu, then choose the option for Root or Zero.
3. The calculator will ask you for a Left Bound and a Right Bound. Move the cursor to the left of the desired zero, select enter, then move it to the right, and select enter again. This tells the calculator *which* zero to look for.
4. The calculator displays the coordinates of the zero, \((x, 0)\).

Did you know?
For complicated functions (like unfamiliar ones mentioned in the syllabus), finding zeros algebraically might be impossible at this stage. The GDC is your lifeline!

4. Finding Local Maxima or Local Minima (Turning Points)

These points are the peaks (maximum) and valleys (minimum) of the graph. They represent the highest or lowest values the function reaches in a specific area (hence "local").

If the function is a quadratic (\(y = ax^2 + bx + c\)), the maximum or minimum point is called the Vertex (see point 6).

GDC Steps (Generic):

1. Graph the function.
2. Select the G-SOLVE or CALC menu, then choose Maximum or Minimum.
3. Just like finding zeros, you define a Left Bound and Right Bound around the peak or valley you are interested in.
4. The calculator displays the coordinates of the turning point, \((x, y)\).

Common Mistake to Avoid:
A local minimum is not necessarily the lowest point on the *entire* graph. It is only the lowest point *relative* to the points immediately surrounding it. Look carefully at the graph's overall behavior!

5. Finding the Intersection of the Graphs of Functions

Finding the intersection points of two graphs is the same as solving simultaneous equations graphically.

When two functions, \(f(x)\) and \(g(x)\), intersect, they share the same \((x, y)\) coordinates at that point.

GDC Steps (Generic for two functions \(Y_1\) and \(Y_2\)):

1. Enter the first function into \(Y_1\) and the second function into \(Y_2\).
2. Ensure the intersection point is visible in your window settings.
3. Select the G-SOLVE or CALC menu, then choose Intersection (or INTERSECT).
4. The calculator will typically ask you to confirm the two curves it is analyzing.
5. It will display the coordinates \((x, y)\) of the intersection point. If there is more than one intersection, you may need to repeat the process or use the cursor to select the point you want.

6. Finding the Vertex of a Quadratic

The vertex is the specific name for the maximum or minimum turning point of a parabola (a quadratic graph).

While you can find the vertex algebraically (using the formula \(x = \frac{-b}{2a}\) for \(y = ax^2 + bx + c\)), the GDC allows for direct calculation:

• If the parabola opens upward (\(a > 0\)), the vertex is the minimum.
• If the parabola opens downward (\(a < 0\)), the vertex is the maximum.

You simply use the standard Maximum or Minimum function described in step 4, or some calculators may have a dedicated Vertex function within the analysis menu.

Recap Checklist: GDC Skills (C3.2 / E3.2)

You must be fluent and confident in performing all these operations on your specific Graphic Display Calculator model:

• Sketch the graph of any function (showing key features).
• Produce a table of values.
• Plot points accurately using those values.
• Find the zeros (roots/x-intercepts) of a function.
• Find local maxima or local minima (turning points).
• Find the intersection point of two graphs.
• Find the vertex of a quadratic function.

Keep practicing these steps! The more you use your GDC, the faster and more intuitive these powerful mathematical analysis tools will become. Good luck!