Investigation 1 – Graphing a(x-p)(x-q) | 探究一：绘制y = a(x-p)(x-q)图像

📚 Investigation 1 – Graphing a(x-p)(x-q) | 探究一：绘制y = a(x-p)(x-q)图像

In this investigation, we explore the factored form of quadratic functions: y = a(x-p)(x-q). By analysing the roles of the parameters a, p and q, we can sketch accurate parabolas without expanding the expression. This form immediately reveals the x-intercepts and, combined with the symmetry of a parabola, allows the vertex and axis of symmetry to be determined efficiently. Understanding how each parameter influences the graph is a fundamental skill in IB Mathematics and provides a solid basis for solving real-world optimisation and modelling problems.

本探究将深入探讨二次函数的因式分解形式：y = a(x-p)(x-q)。通过分析参数a、p和q的作用，我们无需展开表达式就能准确地绘制抛物线。这种形式直接给出了x轴截距，结合抛物线的对称性，可以快速确定顶点和对称轴。理解每个参数对图像的影响是IB数学中的一项基本技能，也为解决现实世界中的最优化和建模问题奠定了坚实基础。

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1. Understanding the Factored Form | 理解因式分解形式

The expression y = a(x-p)(x-q) is known as the factored or intercept form of a quadratic function. Here, a is a non-zero real number, while p and q represent the x-coordinates where the parabola crosses the x-axis. Because the product a(x-p)(x-q) equals zero precisely when x = p or x = q, these two points are the roots (solutions) of the quadratic equation. This structure makes it exceptionally easy to identify key features of the graph directly from the equation.

表达式 y = a(x-p)(x-q) 被称为二次函数的因式分解形式或截距形式。其中a是一个非零实数，p和q表示抛物线与x轴交点的横坐标。由于当x等于p或q时，乘积a(x-p)(x-q)恰好为零，因此这两点就是二次方程的根（解）。这种结构使得我们可以直接从方程中识别图像的关键特征，极为便捷。

The domains of p and q are all real numbers; the parabola may have two distinct x-intercepts, one repeated intercept (when p = q), or no real intercepts if the factored form arises from complex roots. In this investigation we focus on real p and q, but the logic extends to all cases once you connect the form to the expanded quadratic.

p和q的取值范围是所有实数；抛物线可能有两个不同的x轴截距、一个重复截距（当p = q时），或者如果因式形式来自复数根，则没有实数截距。在本探究中，我们主要研究p和q为实数的情况，但当你将这种形式与展开的二次式联系起来时，这样的逻辑可以延伸到所有情形。

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2. Identifying the x-intercepts | 识别x轴截距

Setting y = 0 yields 0 = a(x-p)(x-q). Since a ≠ 0, the Zero Product Property tells us that x-p = 0 or x-q = 0. Hence the parabola meets the x-axis at the points (p, 0) and (q, 0). If p = q, the graph touches the x-axis at one point, often called a double root. The factored form thus gives an immediate visual reference for positioning the parabola on the coordinate plane.

令y = 0得到0 = a(x-p)(x-q)。因为a ≠ 0，根据零乘积性质可知x-p = 0或x-q = 0。因此抛物线与x轴相交于点(p, 0)和(q, 0)。如果p = q，图像在一点接触x轴，这通常被称为重根。因此，因式分解形式能立即为在坐标平面上定位抛物线提供直观参考。

For example, the function y = 2(x+3)(x-5) has x-intercepts at (-3, 0) and (5, 0). Notice how the signs in the brackets are opposite to the intercept values: (x – (-3)) becomes (x+3). Carefully handling these signs is crucial for accurate graphing.

例如，函数 y = 2(x+3)(x-5) 的x轴截距为(-3, 0)和(5, 0)。请注意括号内的符号与截距值相反：(x – (-3)) 写成了 (x+3)。正确处理这些符号对精确绘图至关重要。

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3. The Role of ‘a’ in Opening Direction | 参数a的作用：开口方向

The coefficient a determines whether the parabola opens upward or downward. If a > 0, the quadratic has a positive leading coefficient when expanded, and the parabola opens upward like a U-shape, possessing a minimum point. If a < 0, the parabola opens downward like an inverted U, possessing a maximum point. The magnitude of a does not affect the x-intercepts but alters the steepness and the y-coordinate of the vertex.

系数a决定了抛物线是向上开口还是向下开口。如果a > 0，二次函数展开后首项系数为正，抛物线向上开口呈U形，存在最小值点。如果a < 0，抛物线向下开口呈倒U形，存在最大值点。a的绝对值大小不改变x轴截距，但会影响图形的陡峭程度和顶点的y坐标。

Consider the case a = 0: the function degenerates to y = 0, a horizontal line. Since a must be non-zero for a true quadratic, we restrict a ∈ ℝ {0}. Visualising the sign of a before any calculation helps you predict the overall shape and the nature of the vertex.

考虑a = 0的情况：函数退化为y = 0，一条水平直线。由于真正的二次函数要求a非零，我们限定a ∈ ℝ {0}。在进行任何计算之前先想象a的符号，有助于你预测整体形状以及顶点的性质。

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4. Finding the Axis of Symmetry | 求对称轴

Every parabola is symmetric about a vertical line passing through its vertex. Because the x-intercepts p and q are equidistant from the axis of symmetry, the axis is exactly halfway between them. Thus the equation of the axis of symmetry is given by the mean of the roots:

每条抛物线都关于一条通过其顶点的竖直线对称。由于x轴截距p和q到对称轴的距离相等，对称轴恰好位于它们的中点。因此，对称轴的方程由两个根的平均值给出：

x = (p + q) / 2

This formula holds for any real p and q, including the case p = q, where the axis is simply x = p. No expansion is needed, which makes this form particularly powerful for quick sketching and analysis.

该公式对所有实数p和q成立，包括p = q的情况，此时对称轴简化为x = p。无需展开表达式，这使得该形式在快速绘图和分析时格外强大。

For instance, with intercepts at 1 and 7, the axis is x = (1+7)/2 = 4. With intercepts at -2 and 6, the axis is x = (-2+6)/2 = 2. The symmetry axis anchors the parabola and halves the coordinate plane into two mirror images.

例如，截距为1和7时，对称轴为x = (1+7)/2 = 4。截距为-2和6时，对称轴为x = (-2+6)/2 = 2。对称轴确定了抛物线的中心线，将坐标平面分成两个镜像部分。

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5. Calculating the Vertex Coordinates | 计算顶点坐标

The vertex lies on the axis of symmetry, so its x-coordinate is x_v = (p+q)/2. To find the y-coordinate, substitute x_v into the original factored equation. After simplification, a compact formula emerges:

顶点位于对称轴上，因此其x坐标为x_v = (p+q)/2。为求y坐标，将x_v代入原始因式方程。简化后，可得到一个紧凑的公式：

y_v = -a(p – q)² / 4

Derivation: y_v = a( (p+q)/2 – p )( (p+q)/2 – q ) = a( (q-p)/2 )( (p-q)/2 ) = a( -(p-q)/2 )( (p-q)/2 ) = -a(p-q)²/4. The square ensures y_v is negative when a > 0 and positive when a < 0, which matches the maximum/minimum behaviour.

推导：y_v = a( (p+q)/2 – p )( (p+q)/2 – q ) = a( (q-p)/2 )( (p-q)/2 ) = a( -(p-q)/2 )( (p-q)/2 ) = -a(p-q)²/4。平方确保了当a > 0时y_v为负值，a < 0时y_v为正值，这与最大值/最小值的行为一致。

Therefore, the vertex is ( (p+q)/2 , -a(p-q)²/4 ). This expression highlights that the vertex’s height depends on a and the squared distance between the roots. When p = q, the vertex lies on the x-axis at (p, 0).

因此，顶点坐标为( (p+q)/2 , -a(p-q)²/4 )。这个表达式表明，顶点的高度取决于a以及两根之差的平方。当p = q时，顶点位于x轴上点(p, 0)处。

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6. Sketching the Graph Step-by-Step | 逐步绘制图像

A systematic approach guarantees an accurate parabola plot. Follow these steps using y = a(x-p)(x-q):

采用系统的方法可以保证准确绘制抛物线。按照以下步骤，利用 y = a(x-p)(x-q)：

• Plot the x-intercepts (p, 0) and (q, 0).
• 绘制x轴截距(p, 0)和(q, 0)。
• Draw the axis of symmetry as a dashed vertical line at x = (p+q)/2.
• 在x = (p+q)/2处画出对称轴，用虚线表示。
• Calculate the vertex using the formula and mark it clearly.
• 计算顶点坐标并清楚标记。
• Determine the opening direction from the sign of a, and imagine the curve passing through the intercepts and vertex.
• 根据a的符号确定开口方向，想象曲线穿过截距和顶点。
• Plot additional points if needed, especially when a has a small absolute value and the parabola appears wide.
• 如有必要，特别当a的绝对值较小导致抛物线较宽时，额外绘制其他点。

Connecting the points with a smooth, continuous curve while respecting symmetry yields the graph. Avoid drawing sharp corners near the vertex – the parabola is a smooth curve. Practice with various a, p and q combinations to gain confidence.

用平滑连续的曲线将点连接起来，同时注意对称性，就能得到图像。避免在顶点附近画出尖角——抛物线是一条平滑曲线。通过练习不同的a、p、q组合来增强信心。

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7. Effect of Changing p and q | 改变p和q的影响

Altering p or q while keeping a constant shifts the intercepts horizontally. Increasing p moves the left intercept to the right (if p < q) or repositions the roots, causing the entire parabola to translate horizontally. The axis of symmetry shifts to the new average (p+q)/2, and the vertex follows accordingly. The shape and vertical stretching remain unchanged, but the parabola's position on the x-axis is completely determined by the roots.

在保持a不变的情况下改变p或q，将水平移动截距。增加p会使左截距向右移动（假如p < q）或重新定位根的位置，导致整个抛物线在水平方向上平移。对称轴移动至新的平均值(p+q)/2，顶点也相应移动。形状和垂直拉伸保持不变，但抛物线在x轴上的位置完全由根决定。

For example, compare y = (x-2)(x-6) and y = (x-2)(x-4). The first has intercepts 2 and 6, axis x=4, vertex (4,-4). The second has intercepts 2 and 4, axis x=3, vertex (3,-1). Changing q from 6 to 4 narrowed the distance between roots, bringing the vertex closer to the x-axis and shifting the overall graph leftwards.

例如，比较y = (x-2)(x-6)和y = (x-2)(x-4)。第一个的截距为2和6，对称轴x=4，顶点(4,-4)。第二个的截距为2和4，对称轴x=3，顶点(3,-1)。将q从6改为4缩小了两根之间的距离，使顶点更靠近x轴，并使整个图形向左移动。

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8. Effect of Changing a on Width | a的变化对宽度的影响

The absolute value |a| controls the vertical stretch (or compression) of the parabola. When |a| > 1, the parabola becomes narrower because the function values grow more rapidly as x moves away from the vertex. When 0 < |a| < 1, the parabola widens, appearing flatter near the vertex. A larger |a| does not affect the x-intercepts or symmetry axis, but it lowers (for a>0) or raises (for a<0) the vertex's depth from the x-axis.

|a|的绝对值控制着抛物线的垂直伸缩（或压缩）。当|a| > 1时，由于函数值随x远离顶点而增长得更快，抛物线变得更窄。当0 < |a| < 1时，抛物线变宽，在顶点附近显得更平缓。较大的|a|不会影响x轴截距或对称轴，但会降低（当a>0时）或升高（当a<0时）顶点相对于x轴的深度。

a value	p=4, q=8 顶点y	形状描述
0.5	-2	较宽、平缓
1	-4	标准宽度
2	-8	明显变窄
-1	4	向下开口，标准宽度

Using the vertex formula -a(p-q)²/4, when p=4, q=8 (difference 4, squared 16), the vertex y becomes -a(16)/4 = -4a. The table confirms that scaling a directly scales the vertex depth, causing the visual ‘sharpness’ of the parabola.

利用顶点公式 -a(p-q)²/4，当p=4, q=8（差为4，平方为16）时，顶点y坐标为 -a(16)/4 = -4a。表格证实，按比例调整a会直接缩放顶点深度，从而改变抛物线视觉上的“尖锐”程度。

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9. Relationship Between Factored and Standard Form | 因式形式与标准形式的关系

Expanding y = a(x-p)(x-q) gives the standard form y = ax² – a(p+q)x + apq. This equivalence is valuable when you need to convert between the two representations. The coefficient of x shows the sum of the roots multiplied by -a, while the constant term is the product multiplied by a. From the standard form, the axis of symmetry can also be found by x = -b/(2a), which must match (p+q)/2 – a useful consistency check.

展开 y = a(x-p)(x-q) 得到标准形式 y = ax² – a(p+q)x + apq。这种等价关系在需要在两种表示之间转换时非常有用。x的系数显示了两根之和乘以-a，常数项则是两根之积乘以a。由标准形式，对称轴也可通过 x = -b/(2a) 求得，这必须与(p+q)/2保持一致——一个有用的一致性检验方法。

For instance, starting with y = 3(x-1)(x+4), expand to 3(x² +3x -4) = 3x² +9x -12. Here, the roots are 1 and -4 (sum -3, product -4), and -b/(2a) = -9/(6) = -1.5, which equals (1+(-4))/2. This link deepens your algebraic understanding and facilitates solving various IB problems.

例如，从 y = 3(x-1)(x+4) 开始，展开为 3(x² +3x -4) = 3x² +9x -12。这里，根为1和-4（和为-3，积为-4），而 -b/(2a) = -9/(6) = -1.5，这等于 (1+(-4))/2。这种联系加深了你的代数理解，并有助于解决各种IB问题。

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10. Graphical Interpretation of Discriminant | 判别式的图像解释

When a quadratic is in factored form with real p and q, the discriminant Δ from the standard form is always non-negative: Δ = a²(p+q)² – 4a·apq = a²(p-q)² ≥ 0. If p = q, the discriminant is zero, and the graph touches the x-axis exactly once. If the factored form had complex roots, p and q would not be real; the expression could not be factored over the reals. Therefore, the factored form inherently tells you the nature of the x-intercepts.

当二次函数为实数p和q的因式形式时，由标准形式得到的判别式Δ始终非负：Δ = a²(p+q)² – 4a·apq = a²(p-q)² ≥ 0。如果p = q，判别式为零，图像恰好接触x轴一次。如果因式形式具有复数根，p和q就不是实数；该表达式无法在实数范围内分解。因此，因式形式本质上就揭示了x轴截距的性质。

This connection reinforces why the factored form is so powerful: you immediately see whether the parabola crosses the x-axis twice, touches it once, or, if no real factorisation exists, must remain entirely above or below the axis. It provides a direct geometric interpretation of the algebraic discriminant.

这种联系进一步说明了为何因式形式如此强大：你可以立即看出抛物线与x轴相交两次、接触一次，或者如果不存在实数因式分解则必然完全位于x轴上方或下方。它为代数的判别式提供了直接的几何解释。

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11. Common Mistakes and Tips | 常见错误与提示

One common error is misreading the signs of p and q in the brackets. Remember, (x – p) implies the intercept is at +p, while (x + p) means the intercept is at -p. Another mistake is forgetting to consider the negative sign when calculating the vertex y-coordinate using the formula y_v = -a(p-q)²/4. Students often omit the minus sign, leading to the vertex being placed on the wrong side of the axis.

一个常见错误是误读括号中p和q的符号。记住，(x – p) 意味着截距在 +p，而 (x + p) 意味着截距在 -p。另一个错误是在使用公式 y_v = -a(p-q)²/4 计算顶点y坐标时忘记考虑负号。学生常常漏掉负号，导致顶点被放在对称轴错误的一侧。

Additionally, when sketching, some learners plot the vertex and intercepts but fail to ensure the curve is symmetric. Always check points on both sides of the vertex manually if uncertain. Finally, do not assume the vertex occurs halfway between the intercepts in the y-direction; it is the x-coordinate that halves the segment, not the y-coordinate.

另外，在绘图时，一些学习者标出了顶点和截距，但未确保曲线对称。如果不确定，务必手动检查顶点两侧的点。最后，不要假设顶点在y轴方向位于截距中间；将线段平分的只是x坐标，而不是y坐标。

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12. Summary and Key Takeaways | 总结与核心要点

The factored form y = a(x-p)(x-q) is a remarkably efficient tool for parabolic analysis. It instantly provides the x-intercepts (p,0) and (q,0). The axis of symmetry is the vertical line x = (p+q)/2. The vertex coordinates are ( (p+q)/2 , -a(p-q)²/4 ). The sign of a controls the opening direction, while |a| governs the steepness. Changing p and q translates the graph horizontally, and altering a stretches it vertically.

因式分解形式 y = a(x-p)(x-q) 是抛物线分析中极为高效的工具。它直接给出x轴截距(p,0)和(q,0)。对称轴为竖直线 x = (p+q)/2。顶点坐标为 ( (p+q)/2 , -a(p-q)²/4 )。a的符号控制开口方向，|a|决定陡峭程度。改变p和q会将图像水平平移，改变a则垂直伸缩。

Mastering this form enables you to move seamlessly between graphs and equations, a vital skill for IB Mathematics. Practice with varied examples, connect it to the standard form, and always verify your sketches using symmetry and the computed vertex. With these techniques, graphing quadratics becomes a logical, step-by-step process rather than a guessing game.

掌握这种形式，你就能在图像与方程之间自如切换，这是IB数学中的一项关键技能。通过多样化的例子进行练习，将其与标准形式联系起来，并始终利用对称性和计算出的顶点来检验草图。掌握了这些技巧，绘制二次函数图像就会成为一个逻辑清晰、按部就班的过程，而非猜谜游戏。

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