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 graph of a system of a linear and a quadratic equation is shown. Which system of equations is represented by the graph?
原库参考解析
抛物线图象与二次建模挑战
An object is kicked from a platform. The equation \( h=-4.9t^2+13t+15 \) 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进阶
The equation y = x2 + 6x + 9 intersects the y-axis on the xy-plane at (0, b). What is the value of b ?
原库参考解析
抛物线图象与二次建模进阶
The equation y = 2x2 + 5x + 2 has a graph on the xy-plane that intersects the y-axis at (0, c ). What is the value of c ?
原库参考解析
抛物线图象与二次建模进阶
The y-intercept of the graph of y = 2x 2+27 is (0, y). What is the value of y ?
原库参考解析
抛物线图象与二次建模进阶
The y-intercept of the graph of \( y=x^2+15 \) in the xy-plane is (0, y). What is the value of y?
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 \( y=6x^2+4x+3 \) in the xy-plane?
原库参考解析
抛物线图象与二次建模挑战
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. According to the model, the object was launched from a height of 0 feet and reached its maximum height of 400 feet 5 seconds after it was launched. Based on the model, what was the height. in feet, of the object 9 seconds after it was launched?
原库参考解析
抛物线图象与二次建模进阶
\( f\left(x\right)=-8x^2+118x \)
The function f models the depth below sea level, in meters, of a certain seal x minutes after the seal dives into the water. What is the best interpretation of \( f\left(2\right)=204 \) in this context?
原库参考解析
抛物线图象与二次建模挑战
A machine launches a baseball from ground level. The baseball reaches a maximum height of 134.56 meters above the ground at 2.9 seconds and hits the ground at 5.8 seconds. Which equation represents the height above ground h, in meters, of the baseball t seconds after it is launched?
原库参考解析
抛物线图象与二次建模进阶
A chemist performed an experiment using a set of beakers that can each be modeled by a right circular cylinder with a height of 53 millimeters. The equation \( y=53πx^{2} \) gives the volume y, in cubic millimeters, of the right circular cylinder that models a beaker in this set, where x is the radius, in millimeters, of the base of the cylinder. Which graph best represents this equation in the xy-plane?
原库参考解析
抛物线图象与二次建模挑战
The graph shows the height above ground, in meters, of a ball x seconds after the ball was launched upward from a platform. Which statement is the best interpretation of the marked point (1.0, 3.9) in this context?
原库参考解析
