Introduction to Graphs
Welcome to the world of Graphs! Think of a graph as a "mathematical picture." While equations tell us the rules of a relationship, a graph shows us the story. In this chapter, we will learn how to sketch different types of curves, find their important features, and see how moving an equation slightly can shift its entire shape. Don't worry if you find some of these shapes confusing at first—by the end of these notes, you'll be able to look at an equation and have a very good idea of what its "picture" looks like.
1. Crossing the Axes (Intercepts)
One of the first things we look for in any graph is where it crosses the grid lines. This helps us anchor our sketch.
The y-intercept: This is where the curve crosses the vertical axis. At this point, the horizontal value \(x\) is always 0. To find it, just put \(x = 0\) into your equation.
The x-intercepts (Roots): These are where the curve crosses the horizontal axis. At these points, the vertical value \(y\) is always 0. Finding these often involves solving an equation (like a quadratic or polynomial).
Example: For the curve \(y = x^2 - 4\), the y-intercept is at \((0, -4)\). Setting \(y = 0\) gives \(x^2 - 4 = 0\), so the x-intercepts are at \(2\) and \(-2\).
Quick Review:
• To find the y-intercept, set \(x = 0\).
• To find the x-intercept, set \(y = 0\).
Key Takeaway: Intersections with the axes are the "landmarks" of your graph. They relate directly to the solutions of equations!
2. Quadratic Graphs (Parabolas)
Quadratic equations like \(y = ax^2 + bx + c\) create a "U" shape (if \(a\) is positive) or an "n" shape (if \(a\) is negative). These are called parabolas.
Completing the Square for Sketching
A very clever trick for quadratics is writing them in the form: \(y = a(x + p)^2 + q\).
Why do we do this? Because it tells us the exact "tip" of the curve, called the turning point or stationary point.
• The Turning Point is located at \((-p, q)\).
• The Line of Symmetry is the vertical line \(x = -p\).
Analogy: Imagine throwing a ball into the air. The path it takes is a parabola. The "Turning Point" is the exact moment the ball stops going up and starts coming down (the maximum height).
Memory Aid: Notice that the \(x\)-coordinate of the turning point is the opposite sign of what is inside the bracket. If you see \((x + 3)^2\), the turning point is at \(x = -3\).
Common Mistake to Avoid: When sketching, make sure your parabola looks like a smooth curve, not a "V" shape. It should be rounded at the bottom or top!
Key Takeaway: Use completing the square to find the turning point \((-p, q)\) and the line of symmetry \(x = -p\).
3. Sketching Polynomials
Polynomials are functions like cubics (\(x^3\)) or quartics (\(x^4\)). When sketching these, we look at the roots (where \(y=0\)).
Repeated Roots
Sometimes a factor in an equation is squared, like \(y = (x - 2)^2(x + 1)\).
• A single root (like \(x + 1\)) means the graph crosses the \(x\)-axis.
• A repeated root (like \((x - 2)^2\)) means the graph just touches the axis and bounces back, like a ball hitting the floor. This is a stationary point.
Did you know? The "highest power" of \(x\) tells you the general shape. A positive \(x^3\) graph starts at the bottom left and ends at the top right. A positive \(x^4\) graph looks like a "W" shape.
Key Takeaway: Look for repeated roots to find where the graph "bounces" on the \(x\)-axis.
4. Reciprocal Graphs
The syllabus requires you to know two specific shapes: \(y = \frac{a}{x}\) and \(y = \frac{a}{x^2}\).
Asymptotes: The "Electric Fence"
These graphs have asymptotes. An asymptote is a line that the graph gets closer and closer to, but never actually touches. It’s like an invisible electric fence.
• For both \(y = \frac{a}{x}\) and \(y = \frac{a}{x^2}\), the y-axis (\(x = 0\)) and the x-axis (\(y = 0\)) are asymptotes.
Distinguishing the two:
• \(y = \frac{a}{x}\): The curves are in opposite corners (Top-Right and Bottom-Left if \(a\) is positive). This represents inverse proportion.
• \(y = \frac{a}{x^2}\): Because \(x^2\) is always positive, the graph stays above the x-axis (Top-Right and Top-Left). This is often called the "volcano" shape.
Key Takeaway: Reciprocal graphs never touch the axes (the asymptotes). \(1/x\) has branches in opposite quadrants; \(1/x^2\) is always on the same side of the \(x\)-axis.
5. Graph Transformations
Sometimes we take a standard graph \(y = f(x)\) and change the equation slightly. This "transforms" the picture. There are four main types you need to know:
Vertical Shifts and Stretches (Outside the Function)
When the change is "outside" the \(f(x)\), it affects the y-coordinates. These are "honest" changes because they do exactly what you expect.
• \(y = f(x) + a\): Translation. Moves the graph up by \(a\) units.
• \(y = af(x)\): Stretch. Stretches the graph vertically by scale factor \(a\). (If \(a\) is negative, it also reflects it over the x-axis).
Horizontal Shifts and Stretches (Inside the Function)
When the change is "inside" the bracket with \(x\), it affects the x-coordinates. These are "backwards" changes—they do the opposite of what you might think!
• \(y = f(x + a)\): Translation. Moves the graph left by \(a\) units. (Yes, adding moves it left!)
• \(y = f(ax)\): Stretch. Stretches the graph horizontally by scale factor \(1/a\). (If you multiply \(x\) by 2, the graph actually gets half as wide).
Memory Aid:
• Outside = Vertical (The "V" in Vertical looks like an Outward arrow).
• Inside = Horizontal (The "H" in Horizontal looks like it’s Inside a box).
• Remember: "Inside is Opposite, Outside is Normal."
Encouraging Phrase: Don't worry if transformations seem tricky at first! Just remember to apply the change to the coordinates one step at a time. If it's inside the bracket, do the opposite to the \(x\). If it's outside, do exactly that to the \(y\).
Key Takeaway:
• \(f(x) + a\) moves it up.
• \(f(x + a)\) moves it left.
• \(af(x)\) stretches it tall.
• \(f(ax)\) squashes it thin.