الأسبوع 3 · مجموعة الأصفار الحقيقية لدوال كثيرات الحدود: درجة الدالة والأصفار جبريًا وبيانيًا وإيجاد معامل مجهول · 3 دروس
3
Zeros read from a graph, and an unknown coefficient found from f(a) = 0
Mathematics Prep 3 Term 1, Unit 2, Lesson 2
GoalRead the zeros of a function from where its curve meets the x-axis, and find an unknown coefficient from a given zero.
New words
x-intercept · a point where the curve meets the x-axis; its x-value is a zerocrossing and touching · a curve can cross the axis or just touch it; each gives one zerounknown coefficient · a letter such as b or k in the rule that we find from other informationsubstitute · replace x by a number to get a value
Explanation
The curve of y = f(x) meets the x-axis exactly where y = 0, that is where f(x) = 0. So the zeros of f are the x-values of the points where the curve crosses or touches the x-axis, and the number of different zeros is the number of such points. A curve that never meets the axis has the empty set of zeros. In the other direction, a zero tells us something about the rule. If a is a zero of f, then f(a) = 0, and when the rule contains an unknown letter this gives an equation for it. Example: if 2 is a zero of x² + bx − 10, then 4 + 2b − 10 = 0, so b = 3.
Worked examples
The curve in the picture crosses the x-axis at −2, 0 and 3. Give the set of zeros. The zeros are the x-values of the crossing points: {−2, 0, 3}. There are 3 different zeros.
Read the zeros of y = x² − 4 from its curve. It crosses the axis at −2 and 2, so the set is {−2, 2}. A curve that only touches the axis, such as y = (x − 2)², has one zero, so its set is {2}.
2 is a zero of f(x) = x² + bx − 10. Find b, the other zero, and f(4). f(2) = 0 gives 4 + 2b − 10 = 0, so b = 3 and f(x) = x² + 3x − 10 = (x + 5)(x − 2). The other zero is −5. Also f(4) = 16 + 12 − 10 = 18.
Class activity
Zero detective. Choose two numbers, say 1 and 6. Build a quadratic with these zeros: (x − 1)(x − 6). Expand it, then check that both numbers give 0. Swap with a friend, who must find the zeros of your function.
Practice
1
A curve crosses the x-axis at x = −4, 1 and 2. Write its set of zeros and the number of different zeros.
{−4, 1, 2}: three different zeros.
2
3 is a zero of f(x) = x² − kx + 6. Find k and the other zero.
9 − 3k + 6 = 0, so k = 5. Then x² − 5x + 6 = (x − 2)(x − 3): other zero 2.
3
The zeros of f(x) = x² + bx + c are −1 and 4. Find b and c.
f(x) = (x + 1)(x − 4) = x² − 3x − 4. So b = −3 and c = −4.
4
2 is a zero of f(x) = 2x² + bx − 6. Find b and f(3).
8 + 2b − 6 = 0, so b = −1. f(3) = 18 − 3 − 6 = 9.
5
The curve y = (x − 2)² touches the x-axis at one point. Write the zeros of (x − 2)² as a set.
