2 Functions and Graphs
2.1 Function notation
A function \(f\) assigns to each input \(x\) in its domain exactly one output \(f(x)\). The set of all outputs is the range.
Example 2.1 (Evaluating a function) Let \(f(x) = 2x^2 - 3x + 1\). Find \(f(-2)\) and \(f(a + 1)\).
Solution.
\[ f(-2) = 2(-2)^2 - 3(-2) + 1 = 8 + 6 + 1 = 15, \]
\[ f(a + 1) = 2(a + 1)^2 - 3(a + 1) + 1 = 2a^2 + a. \]
2.2 The graph of a quadratic function
The graph of \(y = ax^2 + bx + c\) is a parabola. Completing the square gives the vertex form
\[ y = a\left(x + \frac{b}{2a}\right)^2 + \frac{4ac - b^2}{4a}, \]
so the vertex is at \(x = -\dfrac{b}{2a}\) and the axis of symmetry is the vertical line through it.
- If \(a > 0\) the parabola opens upwards and the vertex is a minimum.
- If \(a < 0\) the parabola opens downwards and the vertex is a maximum.
Optimisation questions in the Compulsory Part almost always reduce to finding the vertex of a parabola. Complete the square rather than guessing.
Example 2.2 (Maximising an area) A rectangular pen is built against a straight wall using \(40\) m of fencing on the remaining three sides. Find the largest possible area.
Solution. Let the two sides perpendicular to the wall have length \(x\) m. Then the side parallel to the wall is \(40 - 2x\) m, and the area is
\[ A(x) = x(40 - 2x) = -2x^2 + 40x = -2(x - 10)^2 + 200. \]
Since the coefficient of \(x^2\) is negative, \(A\) is greatest when \(x = 10\), giving a maximum area of \(200 \text{ m}^2\).
2.3 Transformations of graphs
Starting from \(y = f(x)\):
| Transformation | New equation | Effect |
|---|---|---|
| Translate up by \(k\) | \(y = f(x) + k\) | Shift \(k\) units in \(+y\) |
| Translate right \(h\) | \(y = f(x - h)\) | Shift \(h\) units in \(+x\) |
| Reflect in \(x\)-axis | \(y = -f(x)\) | Flip vertically |
| Reflect in \(y\)-axis | \(y = f(-x)\) | Flip horizontally |
| Vertical stretch | \(y = k f(x)\) | Scale \(y\)-values by \(k\) |
2.4 Exercises
- Let \(f(x) = x^2 - 6x + 5\). Write \(f(x)\) in vertex form and state the coordinates of the vertex.
- The graph of \(y = f(x)\) is translated \(3\) units to the left and \(2\) units down. Write the equation of the new graph in terms of \(f\).
- Find the maximum value of \(-3x^2 + 12x - 7\).
Answers
- \(f(x) = (x - 3)^2 - 4\); vertex \((3, -4)\).
- \(y = f(x + 3) - 2\).
- \(5\), attained at \(x = 2\).