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
y=0.25x2-7.5x +80.25
The equation gives the estimated stock price y, in dollars, for a certain company x days after the company went public, where 0≤ x ≤20 . Which statement is the best interpretation of (x, y )=(1,73) in this context?
原库参考解析
抛物线图象与二次建模挑战
\( x+5=11 \)
\( y=3x^2+3 \)
At what point (x, y) 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进阶
An auditorium has seats for 2,600 people. Tickets to attend a show at the auditorium currently cost $8.00. For each $1.00 increase to the ticket price, 100 fewer tickets will be sold. This situation can be modeled by the equation \( y=-100x^2+1,800x+20,800 \), where x represents the increase in ticket price, in dollars, and y represents the revenue, in dollars, from ticket sales. If this equation is graphed in the xy-plane, at what value of x is the maximum of the graph?
原库参考解析
抛物线图象与二次建模进阶
An auditorium has seats for 3,000 people. Tickets to attend a show at the auditorium currently cost $8.00. For each $1.00 increase to the ticket price, 100 fewer tickets will be sold. This situation can be modeled by the equation \( y=-100x^2+2,200x+24,000, \) where x represents the increase in ticket price, in dollars, and y represents the revenue, in dollars, from ticket sales. If this equation is graphed in the xy-plane, at what value of x is the maximum of the graph?
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 130 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 194 feet above the ground. which equation defines f?
原库参考解析
抛物线图象与二次建模进阶
An auditorium has seats for 3200 people. Tickets to attend a show at the auditorium currently cost $8.00. For each $1.00 increase to the ticket price, 100 fewer tickets will be sold. This situation can be modeled by the equation y=-100x2+2400x+25600, where x represents the increase in ticket price, in dollars, and y represents the revenue, in dollars, from ticket sales. If this equation is graphed in the xy-plane, at what value of x is the maximum of the graph?
原库参考解析
抛物线图象与二次建模进阶
An auditorium has seats for 3,200 people. Tickets to attend a show at the auditorium currently cost $8.00. For each $1.00 increase to the ticket price, 100 fewer tickets will be sold. This situation can be modeled by the equation y = -100x2 +2,400x+25,600, where x represents the increase in ticket price, in dollars, and y represents the revenue, in dollars, from ticket sales. If this equation is graphed in the xy-plane, at what value of x is the maximum of the graph?
原库参考解析
抛物线图象与二次建模进阶
An object is launched into the air from ground level.According to a quadratic model,3.8 seconds after the object is launched, it reaches its maximum height of 231.04 feet above ground level. Which equation represents this model, where f(t) is the object's height, in feet, above ground level t seconds after it was launched?
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 $150.00, 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 $2,187.50, 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
\( f\left(x\right)=-8x^2+118x \)
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进阶
The y-intercept of the graph of \( y=x^2+18 \)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进阶
An object is kicked from a platform. The equation \( h=-4.9t^2+5t+9 \) 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
