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
The table shows three values of x and their corresponding values of y. There is a quadratic relationship between x and y. An equation that represents this relationship can be written as \( y=27x^2-bx-18 \), where b is a constant. What is the value of b?
原库参考解析
抛物线图象与二次建模挑战
The table shows three values of x and their corresponding values of y. There is a quadratic relationship between x and y. An equation that represents this relationship can be written as \( y=18x^2-bx-48 \), where b is a constant. What is the value of b?
原库参考解析
抛物线图象与二次建模进阶
\( h(t)=-16t^2+b \)
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 43.56 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?
参考答案
抛物线图象与二次建模进阶
What is the y-intercept of the graph of \( y=4x^2+3x+8 \) in the xy-plane?
参考答案
抛物线图象与二次建模进阶
What is the y-intercept of the graph of \( y=8x^2+7x+9 \)in the xy-plane?
参考答案
抛物线图象与二次建模挑战
The equation \( y=−4.9(x−9.1)^{2}+12,400 \) gives the estimated height above ground, y, in meters, of a plane, where x is the number of seconds since it started a parabolic maneuver. If this equation is graphed in the xy-plane, which of the following is the best interpretation of the vertex of the graph?
参考答案
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?
参考答案
抛物线图象与二次建模挑战
The table shows three values of x and their corresponding values of y. There is a quadratic relationship between x and y. An equation that represents this relationship can be written as \( y=36x^2-bx-120 \), where b is a constant. What is the value of b?
参考答案
抛物线图象与二次建模进阶
The elastic potential energy E, in newton‑meters, stored in a spring is the product of \( \frac{1}{2} \)the spring constant, in newtons per meter, and the square of the number of meters, x, the spring is stretched from its equilibrium position. If the spring constant for a certain spring is 74 newtons per meter, which equation gives the elastic potential energy, in newton‑meters, stored in this spring in terms of the number of meters the spring is stretched from its equilibrium position?
参考答案
抛物线图象与二次建模挑战
For the quadratic function \( f \), the table shows three values of x and their corresponding values of \( f(x) \). Which equation defines \( f \)?
参考答案
抛物线图象与二次建模进阶
An object is launched from a height of 129 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, 2 seconds after the object is launched, it reaches a maximum height of 193 feet above the ground.
Which equation defines \( f \)?
Which equation defines \( f \)?
参考答案
抛物线图象与二次建模基础
What is the y-intercept of the graph of \( y=x^2+38 \) in the xy-plane?
参考答案
