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Giải bài 6.30 trang 28 SGK Toán 10 – Kết nối tri thức

Với mỗi hàm số dưới đây, hãy vẽ đồ thị, tập tập giá trị, khoảng đồng biến, khoảng nghịch biến của nó:

ܫTổng hợp đề thi học kì 2 lớp 10 tất cả các môn - Kết nối tri thức

Toán - Văn - Anh - Lí - Hóa - Sinh - Sử - Địa...
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Đề bài

Với mỗi hàm số dưới đây, hãy vẽ đồ thị, tập giá trị, khoảng đồng biến, khoảng nghịch biến của nó: a) \(y =  - {x^2} + 6x - 9\) b) \(y =  - {x^2} - 4x + 1\) c) \(y = {x^2} + 4x\) d) \(y = 2{x^2} + 2x + 1.\)

Phương pháp giải - Xem chi tiết

Cho hàm số \(y =  a{x^2} +bx + c\) -   Xác định tọa độ đỉnh \(I(\frac {-b} {a};\frac {-\Delta} {4a})\) -   Trục đối xứng \(x=\frac {-b} {a}\) -   Giao với trục \(Ox,\,\,Oy.\) -   Xác định tập giá trị của hàm số -   Từ đồ thị tìm khoảng đồng biến, nghịch biến của hàm số

Lời giải chi tiết

a) \(y =  - {x^2} + 6x - 9\) Ta có: \(a =  - 1\) nên parabol quay bề lõm xuống dưới. Đỉnh \(I\left( {3;0} \right).\) Trục đối xứng \(x = 3.\) Giao điểm của đồ thị với trục \(Oy\) là: \(A\left( {0; - 9} \right).\) Parabol cắt trục hoành tại \(x = 3.\)

 

Tập giá trị của hàm số là: \(\left( { - \infty ;0} \right].\) Từ đồ thị ta thấy: Hàm số \(y =  - {x^2} + 6x - 9\) đồng biến trên khoảng \(\left( { - \infty ;3} \right)\) và nghịch biến trên khoảng \(\left( {3; + \infty } \right).\) b) \(y =  - {x^2} - 4x + 1\) Ta có: \(a =  - 1\) nên parabol quay bề lõm xuống dưới. Đỉnh \(I\left( { - 2;5} \right).\) Trục đối xứng \(x =  - 2.\) Giao điểm của hàm số với trục \(Oy\) là: \(\left( {0;1} \right).\) Giao điểm của hàm số với trục \(Ox\) là: \(x =  - 2 + \sqrt 5 \) và \(x =  - 2 - \sqrt 5 .\)

 

Tập giá trị của hàm số là: \(\left( { - \infty ;5} \right].\) Từ đồ thị ta thấy: Hàm số \(y =  - {x^2} - 4x + 1\) đồng biến trên khoảng \(\left( { - \infty ; - 2} \right)\) và nghịch biến trên khoảng \(\left( { - 2; + \infty } \right).\) c) \(y = {x^2} + 4x\) Ta có: \(a = 1 > 0\) nên parabol quay bề lõm lên trên. Đỉnh \(I\left( { - 2; - 4} \right).\) Trục đối xứng \(x =  - 2.\) Giao điểm của hàm số với trục \(Oy\) là: \(\left( {0;0} \right).\) Giao điểm của hàm số với trục \(Ox\) là: \(x = 0\) và \(x =  - 4.\)

 

Tập giá trị của hàm số là: \(\left[ { - 4; + \infty } \right).\) Từ đồ thị ta thấy: Hàm số \(y = {x^2} + 4x\) đồng biến trên khoảng \(\left( { - 2; + \infty } \right)\) và nghịch biến trên khoảng \(\left( { - \infty ; - 2} \right).\) d) \(y = 2{x^2} + 2x + 1\) Ta có: \(a = 2 > 0\) nên parabol quay bề lõm lên trên. Đỉnh \(I\left( { - \frac{1}{2};\frac{1}{2}} \right).\) Trục đối xứng \(x =  - \frac{1}{2}.\) giao điểm của hàm số với trục \(Oy\) là: \(\left( {0;1} \right).\) Đồ thị hàm số không có giao điểm với trục \(Ox.\) Lấy điểm \(\left( {1;5} \right)\) thuộc đồ thị hàm số, điểm đối xứng với điểm đó qua trục đối xứng \(x =  - \frac{1}{2}\) là: \(\left( { - 2;5} \right).\)

 

Tập giá trị của hàm số là: \(\left[ {\frac{1}{2}; + \infty } \right).\) Từ đồ thị ta thấy: Hàm số \(y = 2{x^2} + 2x + 1\) đồng biến trên khoảng \(\left( { - \frac{1}{2}; + \infty } \right)\) và nghịch biến trên khoảng \(\left( { - \infty ; - \frac{1}{2}} \right).\)

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Ph/hs Tham Gia Nhóm Để Cập Nhật Điểm Thi, Điểm Chuẩn Miễn Phí

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