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进阶
A quadratic model gives the predicted instantaneous rate of change, in bacteria per hour, of a bacteria population as a function of the population size. According to the model, the predicted instantaneous rate of change is 0 bacteria per hour for a population size of 0, and the maximum predicted instantaneous rate of change is 0.200 bacteria per hour for a population size of 7,000. Based on this model, what is the predicted instantaneous rate of change, in bacteria per hour, of the bacteria population for a population size of 700?
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进阶
\( h\left(t\right)=-16t^2+b \)
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 2,025 feet above the ground when it is dropped at t = 0. Approximately 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进阶
\( f(x)=-16x^2+99 \)
The function \( f \)gives the estimated height, in feet, of an acorn x seconds after the acorn fell from a tree. Based on the function, what is the estimated height, in feet, of the acorn before it fell from the tree?
原库参考解析
抛物线图象与二次建模挑战
What is an x-intercept of the graph of \( y=x^2-25 \) in the xy-plane?
原库参考解析
抛物线图象与二次建模进阶
For the quadratic function \( f \), the table shows three values of \( x \) and their corresponding values of \( f\left(x\right) \). Which equation defines \( f \)?
原库参考解析
抛物线图象与二次建模基础
An object is kicked from a platform. The equation h = —4.9t2+ 7t ＋ 9 represents this situation, where h is the height of the object above the ground, in meters,t seconds after it is kicked. Which number represents the height, in meters, from which the object was kicked?
原库参考解析
抛物线图象与二次建模挑战
| x | f(x) |
| -1 | 10 |
| 0 | 14 |
| 1 | 20 |
For the quadratic function f, the table shows three values of x and their corresponding values of f(x). Which equation defines f?
原库参考解析
抛物线图象与二次建模挑战
The graph of \( y=2x^2+bx+c \) is shown, where b and c are constants. What is the value of bc?
原库参考解析
抛物线图象与二次建模进阶
An egg is thrown from a rooftop.The equation \( h=-4.9t^2+9t+18 \) represents this situation, where h is the height of the egg above the ground, in meters, t seconds after it is thrown. According to the equation, what is the height, in meters, from which the egg was thrown?
原库参考解析
抛物线图象与二次建模挑战
A submersible device is used for ocean research. The function \( g(x)=−\frac{1}{55}(x+19)(x−35) \) gives the depth below the surface of the ocean, in meters, of the submersible device x minutes after collecting a sample, where \( x>0 \). How many minutes after collecting the sample did it take for the submersible device to reach the surface of the ocean?
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 seal t minutes after the seal entered the water during a dive. The function estimates that the seal reached its maximum depth of 302.4 meters 6 minutes after it entered the water and then reached the surface of the water 12 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
