函數與圖像：二次函數的模樣

1Summary 重點摘要

Function notation, parabola features and discriminants: the three codes of quadratic graphs.

2Vocabulary cards Vocabulary

3Step by step Steps

1. Step 1: The notation f(x) is substitutionThe notation f(x) means: take the value inside the brackets and put it in place of x on the right-hand side. For f(x) = 2x - 3, we get f(4) = 2 × 4 - 3 = 5 and f(-1) = 2 × (-1) - 3 = -5. For f(x) = x² + 1, finding f(-2) requires writing (-2)² + 1 = 5 with brackets around the negative number, otherwise it is easily miscalculated as -4. Substitution underpins everything about functions; every point (c, f(c)) on the graph comes from this move, so it must become automatic.
2. Step 2: The sign of a decides the openingThe graph of y = ax² + bx + c is a parabola whose opening is decided entirely by the sign of a: when a > 0 it opens upward like a bowl and the vertex is the minimum of the whole function; when a < 0 it opens downward like an umbrella and the vertex is the maximum. In y = -3x² + 2x - 1 we have a = -3 < 0, so the parabola opens downward and the vertex is a maximum point. Also, the larger the absolute value of a, the narrower the parabola, which is useful when comparing two curves.
3. Step 3: The vertex formula and extreme valuesThe x-coordinate of the vertex has a fixed formula, x = -b/(2a); substituting this x into the function gives the y-coordinate. For y = x² - 6x + 5: x = -(-6)/(2 × 1) = 3 and y = 9 - 18 + 5 = -4, so the vertex is (3, -4), which is also the minimum of the whole function. When an application asks for the minimum cost, maximum profit or greatest height, the answer is this y-value of the vertex; after finding it, check the sign of a to state whether it is a maximum or a minimum, not just the coordinates.
4. Step 4: The discriminant predicts x-interceptsThe discriminant Δ = b² - 4ac decides how the parabola meets the x-axis: Δ > 0 gives two intercepts, Δ = 0 gives exactly one (the parabola is tangent to the axis), and Δ < 0 gives none, with the whole curve above or below the axis. For y = x² + 2x + 5, Δ = 2² - 4 × 1 × 5 = -16 < 0, so the graph never touches the x-axis. Computing the discriminant before sketching prevents drawing a picture that contradicts the facts, and it is the first check of any rough graph.
5. Step 5: Rules for transforming graphsBasic parabolas such as y = x² follow fixed rules under transformation: f(x) + k shifts the whole curve up k units, or down when k is negative; f(x - h) shifts right h units, because x must be h larger to give the same output, and this is the rule most often remembered backwards; -f(x) flips the graph vertically and reverses the opening. For example y = (x - 2)² + 1 is y = x² shifted right 2 units and up 1 unit, moving the vertex from the origin to (2, 1), which can be read off directly.
6. Step 6: A three-step framework for describing parabolasDescribe a quadratic graph in a fixed order to cover every scoring point: first the direction of opening, decided by the sign of a; second the vertex and the axis of symmetry, which is the line x = -b/(2a); third the intercepts, with the y-intercept at (0, c) and the number of x-intercepts decided by the discriminant. With all three parts complete, a description question has a full framework, no available mark is missed, and the examiner can see every key point at a glance.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• f(x) is substitution: bracket negatives.
• Opening follows a: positive up, negative down.
• Vertex (-b2a, f(-b2a)) answers all max-min questions.
• Δ predicts intercepts: two, one, none.
• Transformations: +k up, -h right, minus flips.