تحليل التمثيلات البيانية للدوال والعلاقات - كتاب الرياضيات - الصف 12 - الفصل 1 - المملكة العربية السعودية

--- SECTION: تحقق من فهمك --- تحقق من فهمك --- SECTION: 1) استثمار --- 1) استثمار: تمثل الدالة: 0 ≤ d ≤ 20 ,v(d) = 0.002d⁴ - 0.11d³ + 1.77d² - 8.6d + 31,0 تقديرًا لاستثمارات أحد رجال الأعمال في السوق المحلية؛ حيث (d)v قيمة الاستثمارات بملايين الريالات في السنة d. --- SECTION: 1A --- 1A) استعمل التمثيل البياني لتقدير قيمة الاستثمارات في السنة العاشرة. ثم تحقق من إجابتك جبريًا. --- SECTION: 1B --- 1B) استعمل التمثيل البياني لتحديد السنوات التي بلغت فيها قيمة الاستثمارات 30 مليون ريال. ثم تحقق من إجابتك جبريًا. --- SECTION: EMPTY --- لا يقتصر استعمال منحنى الدالة على تقدير قيمها، إذ من الممكن استعماله لإيجاد مجال الدالة ومداها. حيث يُعدّ منحنى الدالة ممتدًا من طرفيه إلا إذا حُدّد بنقطة أو دائرة. --- SECTION: مثال 2 --- مثال 2 --- SECTION: إيجاد المجال والمدى --- إيجاد المجال والمدى --- SECTION: EMPTY --- أوجد مجال الدالة f ومداها باستعمال التمثيل البياني المجاور. --- SECTION: المجال --- المجال: • تدل النقطة عند (10- ,8-) على أن المجال يبدأ عند 8- = x . • تدل الدائرة عند النقطة (4 ,4) على أن 4 = x ليست في مجال f . • يدل السهم على الجهة اليمنى من المنحنى على استمرارية المنحنى من اليمين دون حدود (دون توقف). مما سبق يكون مجال الدالة f هو (4- ,∞) U (4- ,8-] . وباستعمال الصفة المميزة للمجموعة يكون المجال هو {x | x ≥ -8, x ≠ -4, x ∈ R}. --- SECTION: المدى --- المدى: إن أقل قيمة للدالة هي (8-)f أو 10- ، وتزداد قيم (x)f بلا حدود عندما تزداد قيم x ، لذا فإن مدى الدالة f هو [10- ,∞). --- SECTION: إرشادات للدراسة --- إرشادات للدراسة اختبار التدريج المناسب: اختر تدريجًا مناسبًا لكل من المحورين y و x للتمكن من رؤية منحنى الدالة بوضوح. لاحظ اختلاف التمثيل الظاهر للدالة f(x) = x⁴ - 20x³ أدناه. --- SECTION: تحقق من فهمك --- تحقق من فهمك --- SECTION: 2A --- 2A --- SECTION: 2B --- 2B --- SECTION: EMPTY --- وزارة التعليم --- SECTION: EMPTY --- الدرس 1-2 تحليل التمثيلات البيانية للدوال والعلاقات 19 --- VISUAL CONTEXT --- **GRAPH**: قيم الاستثمار Description: A line graph showing the value of investments over years. The x-axis represents 'السنوات' (Years) from 0 to 20. The y-axis represents 'قيمة الاستثمارات (بملايين الريالات)' (Investment Value (in millions of Riyals)) from 0 to 42. The curve shows an initial decrease, then an increase, and then a decrease again. X-axis: السنوات Y-axis: قيمة الاستثمارات (بملايين الريالات) Data: The graph shows the investment value over 20 years. It starts at approximately 31 million Riyals at year 0, decreases to a minimum around year 4-6 (approx 18-20 million), then increases to a maximum around year 14 (approx 33-34 million), and finally decreases to around 6 million Riyals at year 20. Key Values: Initial value: ~31 million at d=0, Minimum value: ~18-20 million at d=4-6, Maximum value: ~33-34 million at d=14, Final value: ~6 million at d=20 Context: This graph visually represents the investment function v(d) and is used to answer questions about investment values at specific years and years for specific investment values. **GRAPH**: y = f(x) Description: A graph of a function y=f(x) on a Cartesian coordinate system. The x-axis ranges from -8 to 8, and the y-axis ranges from -8 to 8. The graph shows a curve with a hole at x=-8, a discontinuity at x=4, and an arrow indicating it extends infinitely to the right. X-axis: x Y-axis: y Data: The function starts with an open circle at (-8, 0). It goes down to a local minimum, then up, then has an open circle at (4, 4). From (4,4) it continues to increase to the right, indicated by an arrow. There is a point (0, -4) and (4, 4) is an open circle. Key Values: Open circle at (-8, 0), Open circle at (4, 4), Arrow indicating extension to positive infinity on x-axis, Local minimum around x=-4, y=-8 Context: This graph is used to determine the domain and range of the function f(x) based on its visual representation, including open circles and arrows. **GRAPH**: f(x) = x⁴ - 20x³ Description: A graph of the function f(x) = x⁴ - 20x³ displayed on a calculator screen. The x-axis ranges from -10 to 10, and the y-axis ranges from -10 to 10. This view shows a small portion of the curve, making it appear relatively flat. X-axis: x Y-axis: y Data: The graph shows a curve that appears to be near the x-axis, with a slight dip and rise, within the -10 to 10 range for both axes. It doesn't reveal the full behavior of the function. Key Values: x-range: [-10, 10], y-range: [-10, 10] Context: This graph illustrates how an inappropriate viewing window (scale) can obscure the true behavior of a function, making it seem different from its actual shape. (Note: Some details are estimated) **GRAPH**: f(x) = x⁴ - 20x³ Description: A graph of the function f(x) = x⁴ - 20x³ displayed on a calculator screen with an appropriate viewing window. The x-axis ranges from -5 to 25, and the y-axis ranges from -20000 to 15000. This view clearly shows the local minimum and the overall shape of the quartic function. X-axis: x Y-axis: y Data: The graph shows a quartic function with a local minimum around x=15 and y=-15000. It starts high, dips down to the minimum, and then rises again. The curve passes through the origin (0,0). Key Values: x-range: [-5, 25], y-range: [-20000, 15000], Local minimum around (15, -15000) Context: This graph demonstrates the importance of selecting an appropriate viewing window to accurately represent the behavior and key features of a function, contrasting with the previous zoomed-in view. (Note: Some details are estimated) **GRAPH**: y = g(x) Description: A graph of a semi-circle function y=g(x) on a Cartesian coordinate system. The x-axis ranges from -8 to 8, and the y-axis ranges from -4 to 8. The graph shows an upper semi-circle centered at the origin, extending from x=-4 to x=4. X-axis: x Y-axis: y Data: The function starts at (-4, 0) and ends at (4, 0), forming the upper half of a circle. The maximum y-value is 4 at x=0. The points (-4,0) and (4,0) are closed circles. Key Values: Domain: [-4, 4], Range: [0, 4], Center: (0,0), Radius: 4 Context: This graph is part of an exercise to determine the domain and range of the function g(x). **GRAPH**: y = g(x) Description: A graph of a piecewise linear function y=g(x) on a Cartesian coordinate system. The x-axis ranges from -8 to 8, and the y-axis ranges from -8 to 8. The graph consists of two linear segments. X-axis: x Y-axis: y Data: The first segment is a horizontal line from (-8, 4) to (0, 4), with (-8,4) being an open circle and (0,4) being a closed circle. The second segment is a line starting from (0, 4) (closed circle) and extending downwards to the right, passing through (4, 0) and (8, -4), indicated by an arrow. Key Values: Open circle at (-8, 4), Closed circle at (0, 4), Line segment from (0,4) to (8,-4) and beyond, Arrow indicating extension to positive infinity on x-axis and negative infinity on y-axis Context: This graph is part of an exercise to determine the domain and range of the function g(x).