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
\( x+7=13 \)
\( y=5x^2+5 \)
At what point \( \left(x,\ y\right) \) do the graphs of the equations in the given system intersect?
原库参考解析
抛物线图象与二次建模挑战
An object is launched from a height of 121 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 185 feet above the ground. Which equation defines \( f \) ?
原库参考解析
抛物线图象与二次建模挑战
A chemist performed an experiment using a set of beakers that can each be modeled by a right circular cylinder with a height of 58 millimeters. The equation \( y=58πx^{2} \) 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
An object is launched from a height of 124 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 268 feet above the ground. Which equation defines ?
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 40.96 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
\( f\left(x\right)=-0.0038400x^2+15.236x-15,103 \)
The model shown gives the predicted average Arctic sea ice area \( f(x) \) , in millions of square kilometers, for September of each year x, where x represents a year and \( 1979\le x\le2012 \) . Based on the model, what is the positive difference, in millions of square kilometers, between the predicted average Arctic sea ice area for September of the year 1991 and the predicted average Arctic sea ice area for September of the year 1992? (Round your answer to the nearest thousandth.)
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
A company developed a plan to set the selling price of a product. The company determined that for a selling price of $120.00, zero products would be sold. For each $2.50 decrease in the selling price, the number of products sold would increase by one. For a revenue of exactly $1,437.50, which of the following could be the number of products sold? (revenue = price × number of products sold)
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 139 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 203 feet above the ground. Which equation defines \( f \)?
原库参考解析
抛物线图象与二次建模挑战
A company developed a plan to set the selling price of a product. The company determined that for a selling price of $90.00, zero products would be sold. For each $1.50 decrease in the selling price, the number of products sold would increase by one. For a revenue of exactly $1,336.50, which of the following could be the number of products sold?
(revenue = price × number of products sold)
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. According to the model, the object was launched from a height of 0 feet and reached its maximum height of 1,600 feet 10 seconds after it was launched. Based on the model, what was the height, in feet, of the object 13 seconds after it was launched?
原库参考解析
抛物线图象与二次建模进阶
\( f\left(x\right)=-16x^2+76 \)
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
