More about graphs of functions - Mathematics - HKDSE

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More about Graphs of Functions: Your Visual Guide to Mathematics!

Hey everyone! Welcome to "More about Graphs of Functions". Don't let the name intimidate you. Think of this chapter as learning to read a new language – the language of graphs. Graphs are like pictures that tell the story of a mathematical relationship.

In this chapter, we'll explore the personalities of different graphs, learn how to use them to solve problems without complicated algebra, and even discover how to shift, stretch, and flip them around! It's a super useful skill that makes abstract ideas much easier to see and understand. Let's get started!

Part 1: A Quick Tour of Different Function Graphs

Before we dive deep, let's meet the main characters – the different types of graphs you'll encounter. Each one has its own unique shape and features.

Constant Functions: The Flat Road

The simplest of them all!
Equation: $$y = c$$ (where c is just a number)
Shape: A perfectly horizontal straight line.
Example: The graph of $$y = 3$$ is a horizontal line where every single point has a y-coordinate of 3. It's like walking on perfectly flat ground.

Linear Functions: The Consistent Slope

You've seen these many times before!
Equation: $$y = mx + c$$
Shape: A straight line that slopes upwards or downwards.
Example: The graph of $$y = 2x + 1$$ is a straight line. For every step you take to the right (increase x by 1), you go up two steps (y increases by 2). It's like walking up a ramp with a constant steepness.

Quadratic Functions: The U-Turn

These are the curvy 'U' shapes.
Equation: $$y = ax^2 + bx + c$$
Shape: A parabola, which can open upwards (like a smile 😊) or downwards (like a frown ☹️).
Example: The path of a ball thrown into the air is a parabola. It goes up, reaches a highest point (the vertex), and comes back down.

Trigonometric Functions: The Never-Ending Wave

These functions describe repeating patterns.
Equations: $$y = \sin x$$ and $$y = \cos x$$
Shape: A smooth, continuous wave that goes on forever.
Example: Think about sound waves, ripples in a pond, or a swinging pendulum. These all follow a repeating, wave-like pattern, just like a sine or cosine graph.

A Quick Note for Everyone

You might also see Exponential and Logarithmic functions. While a deep dive into their graphs is usually for students taking the non-foundation topics, it's cool to know what they look like! Exponential functions show rapid growth (like a "J" curve), and logarithmic functions show growth that slows down.

Comparing Graphs: What to Look For?

When you're asked to compare graphs, here are the four key features to talk about:

Domain: What are the possible x-values? For linear and quadratic functions, x can be any real number. For some other functions, there might be restrictions.
Maximum / Minimum Values: Is there a highest point (maximum) or a lowest point (minimum)? A smiling parabola has a minimum value at its vertex. A frowning one has a maximum. Sine and cosine waves have both!
Symmetry: Is the graph a mirror image of itself? Parabolas have a line of symmetry running through their vertex.
Periodicity: Does the graph repeat itself in a regular cycle? This is the key feature of trigonometric graphs. The length of one full cycle is called the period.

Key Takeaway for Part 1

Every type of function has a characteristic graph shape. By looking at its shape, we can understand its key properties like its highest/lowest points, symmetry, and whether it repeats.

Part 2: Solving Problems with Pictures: Graphical Solutions

Did you know you can solve equations just by looking at a graph? It's a powerful visual method that can make things much clearer than just staring at symbols and numbers.

Solving $$f(x) = k$$: Where Do the Graphs Meet?

Solving an equation like $$x^2 - x - 5 = 1$$ can be tricky. But what if we thought about it graphically?

The equation $$f(x) = k$$ is really asking: "For the function $$y = f(x)$$, at which x-values is the y-value equal to $$k$$?"

This is the same as finding the intersection points of two graphs:

The graph of $$y = f(x)$$ (this could be a curve, a wave, etc.)
The graph of $$y = k$$ (this is always a horizontal line!)

The x-coordinates of where these two graphs cross are the solutions to the equation!

Step-by-Step Guide:

Let's solve $$x^2 - 3 = 1$$ using the graph of $$y = x^2 - 3$$.
Step 1: You are given the graph of $$y = x^2 - 3$$ (a parabola).
Step 2: On the same axes, draw the horizontal line $$y = 1$$.
Step 3: Look for where the parabola and the line intersect. You'll see they meet at two points.
Step 4: Read the x-coordinates of these intersection points. They are at $$x = -2$$ and $$x = 2$$.
Result: The solutions are $$x = -2$$ and $$x = 2$$. That's it! No factoring needed.

Solving Inequalities like $$f(x) > k$$: Who's On Top?

We can use the same idea to solve inequalities. It's all about comparing the position of the curve ($$y = f(x)$$) to the horizontal line ($$y = k$$).

To solve $$f(x) > k$$, look for the range of x-values where the curve is above the line $$y=k$$.
To solve $$f(x) < k$$, look for the range of x-values where the curve is below the line $$y=k$$.

Memory Aid: Think of the ">" symbol as an arrow pointing up (above) and the "<" symbol as an arrow pointing down (below).

Quick Review Box

$$f(x) = k$$: Where the graph of $$f(x)$$ INTERSECTS the line $$y=k$$.
$$f(x) > k$$: Where the graph of $$f(x)$$ IS ABOVE the line $$y=k$$.
$$f(x) < k$$: Where the graph of $$f(x)$$ IS BELOW the line $$y=k$$.
(For $$\geq$$ or $$\leq$$, just include the intersection points in your solution!)

Common Mistakes Alert!

When you solve an inequality, your answer should be a range of x-values, not y-values. For example, an answer should look like "$$x > 5$$" or "$$ -1 < x < 3$$", NOT "$$y > k$$". You are finding the cause (x) of the condition, not stating the condition itself.

Key Takeaway for Part 2

We can solve equations and inequalities by drawing the function's graph and a horizontal line. The solutions are the x-values where the graph intersects, is above, or is below the line.

Part 3: Graph Makeovers: Transformations of Functions

Imagine you have a graph of a function, say $$y = f(x)$$. Transformations are simple rules that let you move, stretch, shrink, and flip that graph to create a new one, without having to plot all the points from scratch!

Don't worry if this seems tricky at first, we'll break it down with simple analogies.

1. Vertical Shift ($$y = f(x) + k$$) - The Elevator Ride

This is the easiest one! It moves the whole graph straight up or down.

If you have $$y = f(x) + k$$ (with k > 0), the graph shifts UP by k units.
If you have $$y = f(x) - k$$ (with k > 0), the graph shifts DOWN by k units.

Analogy: Imagine your graph is in an elevator. Adding k outside the function is like pressing the "up" button. Subtracting k is like pressing "down". The shape doesn't change, just its vertical position.

2. Horizontal Shift ($$y = f(x+k)$$) - The Sidestep

This one moves the graph left or right. Be careful, it's a bit counter-intuitive!

If you have $$y = f(x + k)$$ (with k > 0), the graph shifts to the LEFT by k units.
If you have $$y = f(x - k)$$ (with k > 0), the graph shifts to the RIGHT by k units.

Memory Trick: Think of anything inside the bracket with 'x' as being in "Opposite Land". So "+k" inside means you do the opposite and move left (in the negative direction). "-k" means you move right.

3. Vertical Stretch/Reflection ($$y = kf(x)$$) - The Slinky

This stretches or squishes the graph vertically. The change happens outside the function, so it affects the y-values.

If $$|k| > 1$$, the graph is stretched vertically away from the x-axis. (It gets taller/steeper).
If $$0 < |k| < 1$$, the graph is compressed vertically towards the x-axis. (It gets shorter/flatter).
If $$k$$ is negative, the graph is also reflected across the x-axis (flipped upside down).

Analogy: Imagine the graph is a Slinky toy. Multiplying by a big number stretches it out. Multiplying by a fraction squishes it. A negative sign flips it over.

4. Horizontal Stretch/Reflection ($$y = f(kx)$$) - The Accordion

This stretches or squishes the graph horizontally. The change is inside the bracket, so it's back to "Opposite Land"!

If $$|k| > 1$$, the graph is compressed horizontally towards the y-axis. (It gets narrower).
If $$0 < |k| < 1$$, the graph is stretched horizontally away from the y-axis. (It gets wider).
If $$k$$ is negative, the graph is also reflected across the y-axis (flipped sideways).

Analogy: This is like playing an accordion. A bigger 'k' squishes it together, while a smaller fractional 'k' pulls it apart.

Putting It All Together

Often, you'll see transformations combined. For example, how do we get the graph of $$y = -(x-2)^2 + 3$$? We can build it step-by-step from the basic graph of $$y = x^2$$.

Start with $$y = x^2$$ (our basic U-shape parabola).
Deal with the horizontal shift: The `(x-2)` part means we shift the graph RIGHT by 2 units. Now we have the graph of $$y = (x-2)^2$$.
Deal with the reflection: The `-` sign out front means we reflect the graph in the x-axis. Now we have $$y = -(x-2)^2$$.
Deal with the vertical shift: The `+3` at the end means we shift the graph UP by 3 units. We get our final graph, $$y = -(x-2)^2 + 3$$.

Key Takeaway for Part 3

Changes outside the function (e.g., $$f(x)+k$$, $$k f(x)$$) affect the graph vertically and are intuitive. Changes inside the function with x (e.g., $$f(x+k)$$, $$f(kx)$$) affect the graph horizontally and are often the opposite of what you'd expect.