Functions and Graphs: Your Ultimate Study Guide!
Hey everyone! Welcome to your friendly guide to Functions and Graphs. Don't worry if this topic sounds a bit scary. We're going to break it all down into simple, easy-to-understand pieces. Functions are one of the most important ideas in mathematics because they help us describe relationships in the world around us, from the path of a basketball to how your phone bill is calculated. Ready? Let's get started!
1. What on Earth is a Function?
Think of a function like a magical vending machine. You put something in (an input), the machine follows a specific rule, and it gives you something out (an output).
Analogy: The Vending Machine
- Input: You press the button 'B4'.
- Rule: The machine knows 'B4' means 'a packet of crisps'.
- Output: You get a packet of crisps.
The most important rule for a function is: For every one input, there can only be ONE output. If you pressed 'B4' and sometimes got crisps and other times got a chocolate bar, the machine would be broken. It wouldn't be a function!
Key Words You Need to Know
- Independent Variable: This is your input. It's the value you choose to put into the function. We usually call it x.
- Dependent Variable: This is your output. Its value depends on the input you chose. We usually call it y or f(x).
- Domain: The set of ALL possible inputs (all the x-values you are allowed to use).
- Co-domain: The set of ALL possible outputs.
Function Notation: f(x)
We often write functions using a special notation, like f(x). You say it as "f of x". It's just a fancy way of saying "the output when the input is x". So, y and f(x) mean the same thing!
Example: Let's say our function rule is "double the input and add 1".
In math language, we write this as: $$f(x) = 2x + 1$$
- If our input is x = 3, we find the output: $$f(3) = 2(3) + 1 = 6 + 1 = 7$$
- If our input is x = -5, we find the output: $$f(-5) = 2(-5) + 1 = -10 + 1 = -9$$
Three Ways to Show a Function
You can represent the same function in different ways:
1. Tabular Method (a table):
This is great for seeing specific input-output pairs. For $$f(x) = 2x + 1$$:
Input (x) | Output (f(x))
-1 | -1
0 | 1
1 | 3
2 | 5
2. Algebraic Method (an equation):
This is the rule itself. It's powerful because it works for any input.
Example: $$f(x) = 2x + 1$$
3. Graphical Method (a graph):
This gives you a picture of the function, showing the relationship between all the inputs and outputs. For a linear function like $$f(x) = 2x + 1$$, the graph is a straight line.
Key Takeaway for Section 1
A function is a rule that takes an input (x) and gives exactly one output (y or f(x)). You can show it as a table, an equation, or a graph.
2. The Mighty Quadratic Function: All About Parabolas
You've seen these before! A quadratic function has an $$x^2$$ term, and its graph is a beautiful U-shaped curve called a parabola.
The standard form is: $$y = ax^2 + bx + c$$ (where 'a' cannot be zero).
Let's decode the graph's secrets by looking at a, b, and c!
Features of a Parabola
1. Direction of Opening:
This is the easiest one! It's all about the value of 'a'.
- If a > 0 (positive), the parabola opens upwards. (Think: a is positive, so it's a happy face :) )
- If a < 0 (negative), the parabola opens downwards. (Think: a is negative, so it's a sad face :( )
2. The y-intercept:
This is where the graph crosses the y-axis. At this point, x is always 0.
If $$y = ax^2 + bx + c$$, and you put $$x=0$$, you get $$y = a(0)^2 + b(0) + c = c$$.
So, the y-intercept is always (0, c). Super easy!
3. The Vertex:
This is the turning point of the parabola.
- If the parabola opens upwards, the vertex is the lowest point (a minimum).
- If the parabola opens downwards, the vertex is the highest point (a maximum).
4. The Axis of Symmetry:
This is a vertical line that cuts the parabola into two perfect mirror images. It passes right through the vertex.
The equation of this line is $$x = -b / (2a)$$. This formula is super useful because the x-coordinate of the vertex is also $$-b / (2a)$$!
5. The x-intercepts (or Roots):
These are the points where the graph crosses the x-axis. At these points, y is always 0. So, we are solving the equation $$ax^2 + bx + c = 0$$.
How many x-intercepts are there? We can use the discriminant ($$\Delta = b^2 - 4ac$$) to find out!
- If $$\Delta > 0$$, there are two distinct x-intercepts. (The graph crosses the x-axis twice).
- If $$\Delta = 0$$, there is one x-intercept. (The vertex touches the x-axis).
- If $$\Delta < 0$$, there are no real x-intercepts. (The graph never touches the x-axis).
Key Takeaway for Section 2
The graph of $$y = ax^2 + bx + c$$ is a parabola. The sign of 'a' tells you if it opens up or down. 'c' gives the y-intercept. The axis of symmetry is $$x = -b / (2a)$$, which also gives the x-coordinate of the vertex. The discriminant tells you how many times the graph hits the x-axis.
3. Finding Maximum and Minimum Values
As we saw, the vertex of a parabola is either its highest point (maximum) or lowest point (minimum). The 'value' of the function at this point is simply the y-coordinate of the vertex.
Method 1: From a Graph (For Everyone)
If you are given the graph, this is the easiest job in the world. 1. Find the vertex (the turning point). 2. Read its y-coordinate. 3. If the parabola opens up, that's your minimum value. 4. If the parabola opens down, that's your maximum value.
Method 2: By Algebra (Non-foundation Topic)
How do you find the vertex if you only have the equation? You have two great options.
Option A: Use the Axis of Symmetry Formula
This is usually the quickest way!
1. Find the x-coordinate of the vertex using the formula: $$x = -b / (2a)$$
2. Substitute this x-value back into the original function $$y = ax^2 + bx + c$$ to find the corresponding y-value.
3. This (x, y) pair is your vertex! The y-value is your max/min value.
Example: Find the minimum value of $$y = 2x^2 - 8x + 5$$
- Here, a = 2, b = -8, c = 5. 'a' is positive, so it's a minimum.
- Step 1: x-coordinate of vertex = $$-(-8) / (2 * 2) = 8 / 4 = 2$$.
- Step 2: Sub x=2 back in: $$y = 2(2)^2 - 8(2) + 5 = 2(4) - 16 + 5 = 8 - 16 + 5 = -3$$.
- Answer: The vertex is (2, -3). The minimum value of the function is -3.
Option B: Completing the Square
This method changes the form of the equation from $$y = ax^2 + bx + c$$ to the vertex form $$y = a(x - h)^2 + k$$. Once it's in this form, the vertex is simply (h, k).
Don't worry if this seems tricky at first, it just takes practice!
Key Takeaway for Section 3
The maximum or minimum value of a quadratic function is the y-coordinate of its vertex. You can find it by looking at the graph or by using the formula $$x = -b / (2a)$$ to find the vertex algebraically.
4. Solving Equations and Inequalities with Graphs
Graphs are not just pretty pictures; they are powerful tools for solving problems! You can find solutions just by looking at where lines and curves cross.
Solving f(x) = k
Solving an equation like $$x^2 - 2x - 2 = 1$$ graphically means finding the x-values that make it true.
Step-by-step:
1. Think of it as two separate graphs: $$y = f(x)$$ (the curve) and $$y = k$$ (a horizontal line).
2. On the same axes, draw the graph of $$y = f(x)$$ (e.g., $$y = x^2 - 2x - 2$$) and the line $$y = k$$ (e.g., $$y = 1$$).
3. The solutions are the x-coordinates of the points of intersection!
Example: Using the graph of $$y = x^2 - 2x - 2$$, solve $$x^2 - 2x - 2 = 1$$.
You would draw the parabola and then draw the horizontal line y=1. If they cross at x = -1 and x = 3, then those are your solutions.
Solving f(x) > k and f(x) < k
This is about finding a range of x-values, not just specific points.
- To solve f(x) > k, you are looking for all the x-values where the graph of $$y = f(x)$$ is ABOVE the line $$y = k$$.
- To solve f(x) < k, you are looking for all the x-values where the graph of $$y = f(x)$$ is BELOW the line $$y = k$$.
Memory Aid: Think of '>' as 'greater than' or 'higher than' (above). Think of '<' as 'less than' or 'lower than' (below).
Key Takeaway for Section 4
To solve $$f(x) = k$$ graphically, find where the graph $$y=f(x)$$ intersects the line $$y=k$$. To solve $$f(x) > k$$, find where the graph is above the line. To solve $$f(x) < k$$, find where the graph is below the line.
5. Transformations of Graphs
Transformations are ways to move, stretch, or flip a graph. If you know the graph of a basic function like $$y = f(x)$$, you can sketch new, related graphs without making a new table of values.
The Four Basic Transformations
Let's use a base function, $$y = f(x)$$.
1. Vertical Shift: $$y = f(x) + k$$ (Moves Up/Down)
- If k is positive, the graph shifts UP by k units.
- If k is negative, the graph shifts DOWN by k units.
This one is intuitive and does exactly what you'd expect.
2. Horizontal Shift: $$y = f(x + k)$$ (Moves Left/Right)
- If k is positive (e.g., $$f(x+3)$$), the graph shifts LEFT by k units.
- If k is negative (e.g., $$f(x-3)$$), the graph shifts RIGHT by k units.
Warning: Common Mistake! This one is the opposite of what you might expect. Remember: x lies to you! Adding to x moves it in the negative direction (left), and subtracting from x moves it in the positive direction (right).
3. Vertical Stretch/Compression: $$y = kf(x)$$ (Stretches/Squeezes Vertically)
- If $$|k| > 1$$, the graph is stretched vertically (it gets taller/skinnier).
- If $$0 < |k| < 1$$, the graph is compressed vertically (it gets shorter/wider).
- If k is negative, the graph is also reflected across the x-axis (flipped upside down).
4. Horizontal Stretch/Compression: $$y = f(kx)$$ (Stretches/Squeezes Horizontally)
- If $$|k| > 1$$, the graph is compressed horizontally by a factor of 1/k.
- If $$0 < |k| < 1$$, the graph is stretched horizontally by a factor of 1/k.
- If k is negative, the graph is also reflected across the y-axis (flipped sideways).
This one is also counter-intuitive, like the horizontal shift. A big 'k' inside the bracket squishes the graph horizontally.
Did you know?
The beautiful parabolic shape you see in a quadratic graph is found all over nature and engineering! The path of a thrown object, the shape of a satellite dish, and the cables on a suspension bridge are all parabolas.
Key Takeaway for Section 5
Changes *outside* the function bracket (like $$f(x)+k$$ and $$kf(x)$$) affect the graph vertically. Changes *inside* the function bracket (like $$f(x+k)$$ and $$f(kx)$$) affect the graph horizontally and often in a counter-intuitive way.