这部分需要会什么？
- 从顶点式读取顶点和对称轴
- 求二次函数的最大值或最小值
- 处理定义域受限时的最值
核心概念与公式
\(y=a(x-h)^2+k\) 的顶点为 \((h,k)\)，对称轴是 \(x=h\)。\(a>0\) 开口向上，全体实数上最小值为 \(k\)；\(a<0\) 开口向下，全体实数上最大值为 \(k\)。
对展开式 \(ax^2+bx+c\)，顶点横坐标 \(h=-b/(2a)\)，纵坐标为 \(f(h)\)。若输入仅限一个区间，先确认顶点是否在区间内，并比较区间端点；整数输入还需检查顶点附近的允许整数。
做题时怎样使用
- 先判断开口方向与实际输入范围
- 从顶点式读顶点，或用 \(-b/(2a)\) 求横坐标
- 区分“最值是多少”和“何时达到最值”，必要时检查端点
容易混淆的地方
\((x+4)^2\) 对应顶点横坐标 \(-4\)，不是 4。顶点横坐标表示位置，纵坐标才是函数值。
用一个例题串起方法
知识点教学例题\(h(t)=-5(t-3)^2+46\)，其中 \(0\le t\le6\)。最大高度是多少，何时达到？
展开解题思路 ＋
平方项非负，乘以 \(-5\) 后不大于零。\(t=3\) 在允许范围内，此时 \(h=46\)，所以最大高度为 46，在 \(t=3\) 时达到。
本知识点题库
先独立作答，再查看答案与解析。
顶点・对称轴与最值进阶
\( y=x^2-18x+51 \)
The given equation relates the variables x and y. For what value of x does the value of y reach its minimum?
原库参考解析
顶点・对称轴与最值进阶
\( g\left(x\right)=5\left(x^2-14x+49\right)+k \)
The function g is defined by the given equation, where k is a constant. For what value of x does \( g\left(x\right) \) reach its minimum?
原库参考解析
顶点・对称轴与最值挑战
\( f(x)=18x^{2}−45x+16 \)
The given equation defines the function \( f \) f . For what value of x does f (x) reach its minimum?
原库参考解析
顶点・对称轴与最值挑战
\( f\left(x\right)=ax^2+3x+c \)
In the given quadratic function, a and c are constants. The graph of \( y=f\left(x\right) \) in the xy-plane is a parabola that opens upward and has a vertex at the point (h, k), where h and k are constants. If k < 0 and \( f\left(-7\right)=f\left(1\right) \), which of the following must be true?
Ⅰ. c < 0
Ⅱ. a ≥ 1
原库参考解析
顶点・对称轴与最值进阶
\( y=x^2-22x+80 \)
The given equation relates the variables x and y. For what value of x does the value of y reach its minimum?
原库参考解析
顶点・对称轴与最值进阶
\( y=x^2-16x+44 \)
The given equation relates the variables x and y. For what value of x does the value of y reach its minimum?
原库参考解析
顶点・对称轴与最值挑战
Function \( f \) is a quadratic function where \( f\left(-21\right)=0 \) and \( f\left(-5\right)=0 \). The graph of \( y=f\left(x\right) \) in the xy-plane has a vertex at \( \left(r,\ -64\right) \). What is the value of r?
原库参考解析
顶点・对称轴与最值进阶
The function \( f\left(x\right)=\frac{1}{9}\left(x-4\right)^2+5 \) gives a toy car's height above the ground \( f\left(x\right) \), in inches, x seconds after it started moving on an elevated track, where \( 0\le x\le10 \). Which of the following is the best interpretation of the vertex of the graph of \( y=f\left(x\right) \) in the xy-plane?
原库参考解析
顶点・对称轴与最值进阶
\( g\left(x\right)=x^2+\frac{27}{2}x \)
Which expression represents the minimum value of the given function?
原库参考解析
顶点・对称轴与最值进阶
\( r\left(t\right)=53t-2t^2 \)
The function r is defined by the given equation. The function s is defined by \( s\left(t\right)=r\left(t\right)+1 \). Which expression represents the maximum value of \( s\left(t\right) \)?
原库参考解析
顶点・对称轴与最值进阶
\( f\left(x\right)=\left(2x-20\right)\left(2x+8\right) \)
What is the minimum value of the given function?
原库参考解析
顶点・对称轴与最值进阶
The function \( f\left(x\right)=\frac{1}{9}\left(x-6\right)^2+3 \) gives a toy car's height above the ground \( f\left(x\right) \), in inches, x seconds after it started moving on an elevated track, where \( 0\le x\le10 \). Which of the following is the best interpretation of the vertex of the graph of \( y=f\left(x\right) \) in the xy-plane?
原库参考解析
