这部分需要会什么？
- 根据零点、顶点或给定点选用二次模型形式
- 求模型中的比例系数
- 将交点与最值解释为实际情境中的量
核心概念与公式
已知两个零点 \(r,s\)，可写 \(y=a(x-r)(x-s)\)；已知顶点 \((h,k)\)，可写 \(y=a(x-h)^2+k\)。再代入另一个已知点求 \(a\)。若已知三个不同横坐标的点，也可代入 \(ax^2+bx+c\) 建立方程组。
固定等间距输入下，线性模型的一阶差相同，二次模型的二阶差相同。建模后需要说明变量单位、可用范围，以及零点或顶点在题目中的实际意义。
做题时怎样使用
- 根据已知信息选择最容易直接代入的形式
- 用额外的点求系数，不要默认 \(a=1\)
- 算出所求量后，用范围和情境排除不合适的解
容易混淆的地方
同一对零点对应很多不同的抛物线，必须有更多信息确定开口宽度。时间为负的代数解可能不属于实际过程。
用一个例题串起方法
知识点教学例题抛物线与横轴交于 \(x=1\)、\(x=5\)，并经过 \((0,10)\)。求其方程与顶点。
展开解题思路 ＋
设 \(y=a(x-1)(x-5)\)。代入 \((0,10)\) 得 \(5a=10\)，所以 \(a=2\)。对称轴 \(x=(1+5)/2=3\)，顶点纵坐标为 \(2(2)(-2)=-8\)。方程为 \(y=2(x-1)(x-5)\)，顶点为 \((3,-8)\)。
本知识点题库
先独立作答，再查看答案与解析。
抛物线图象与二次建模挑战
A quadratic function gives the estimated length of daylight \( d(t) \), in hours, in a certain city t months after March 1, where \( 0\le t\le7 \) . According to the function, the estimated length of daylight is 12.66 hours 6 months after March 1 and the maximum estimated length of daylight is 13.91 hours 3.5 months after March 1. Based on this function, what is the estimated length of daylight, in hours, on March 1?
原库参考解析
抛物线图象与二次建模进阶
A quadratic function models the height, in feet, of an object above the ground in terms of the time, in seconds, after the object was launched. The model estimates that the object was launched from an initial height of 0.00 feet above the ground and was at a maximum height of 207.36 feet above the ground 3.6 seconds after it was launched. The model also estimates that the object was at a height of 184.32 feet above the ground x seconds after it was launched. What is a possible value of x?
原库参考解析
抛物线图象与二次建模挑战
\( h\left(t\right)=-16t^2+b \)
The function h estimates an object's height, in feet, above the ground t seconds after the object is dropped, where b is a constant. The function estimates that the object is 19.36 feet above the ground when it is dropped at \( t=0 \). How many seconds after being dropped does the function estimate the object will hit the ground?
原库参考解析
抛物线图象与二次建模挑战
\( h(t)=-16t^2+b \)
The function h estimates an object's height, in feet, above the ground t seconds after the object is dropped, where b is a constant. The function estimates that the object is 23.04 feet above the ground when it is dropped at \( t=0 \). How many seconds after being dropped does the function estimate the object will hit the ground?
原库参考解析
抛物线图象与二次建模挑战
The quadratic function a models the depth, in meters, below the surface of the water of a Weddell seal t minutes after the seal entered the water during a dive. The function estimates that the seal reached its maximum depth of 382.2 meters 7 minutes after it entered the water and then reached the surface of the water 14 minutes after it entered the water. Based on the function, what was the estimated depth, to the nearest meter, of the seal 10 minutes after it entered the water?
原库参考解析
抛物线图象与二次建模挑战
The quadratic function g models the depth, in meters, below the surface of the water of a Weddell seal t minutes after the seal entered the water during a dive. The function estimates that the seal reached its maximum depth of 396.8 meters 8 minutes after it entered the water and then reached the surface of the water 16 minutes after it entered the water. Based on the function, what was the estimated depth, to the nearest meter, of the seal 11 minutes after it entered the water?
原库参考解析
抛物线图象与二次建模进阶
The function \( f(x)=-x(x-401) \), where \( 0\le x\le401 \), gives the speed, in meters per second, of a particle x seconds after it started traveling. If the function \( f \) is graphed in the xy-plane, where \( y=f(x) \), which statement is the best interpretation of the point (1, 400) on the graph?
原库参考解析
抛物线图象与二次建模挑战
The graph gives the estimated revenue, in dollars, for a company when the price of the company's product is x dollars. What does the y-value of the vertex of the graph represent in this context?
原库参考解析
抛物线图象与二次建模挑战
What is the y-intercept of the graph of \( y=6x^2+7x+21 \) in the xy-plane?
原库参考解析
抛物线图象与二次建模基础
The graph of a system of a linear equation and a quadratic equation is shown. What is the solution (x, y) to this system?
原库参考解析
抛物线图象与二次建模进阶
An object is launched from a height of 132 feet above the ground. A quadratic function \( f \) models the height above the ground, in feet, of the object t seconds after it is launched. According to the model, 3 seconds after the object is launched, it reaches a maximum height of 276 feet above the ground. Which equation defines ?
原库参考解析
抛物线图象与二次建模挑战
The graph of \( y=6x^2+bx+c \) is shown, where b and c are constants. What is the value of \( bc \)?
原库参考解析
