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
An equation of the graph shown is \( y=−\frac{3}{8} \)\( \left(x-p\right)^2 \) \( +6 \), where p is an integer constant. What is the value of p?
参考答案
顶点・对称轴与最值进阶
\( f(x)=x^2-12x+30 \)
What is the minimum value of the given function?
What is the minimum value of the given function?
参考答案
顶点・对称轴与最值进阶
\( f(x)=x^2-10x+11 \)
What is the minimum value of the given function?
What is the minimum value of the given function?
参考答案
顶点・对称轴与最值进阶
\( f(x)=x^2-20x+76 \)
What is the minimum value of the given function?
What is the minimum value of the given function?
参考答案
顶点・对称轴与最值挑战
\( f(x)=4x^2+56x+197 \)
The function g is defined by \( g(x)=f(x+8) \). What is the minimum value of \( g(x) \)?
The function g is defined by \( g(x)=f(x+8) \). What is the minimum value of \( g(x) \)?
参考答案
顶点・对称轴与最值挑战
\( f(x)=4x^2+72x+325 \)
The function g is defined by \( g(x)=f(x+6) \). What is the minimum value of \( g(x) \)?
The function g is defined by \( g(x)=f(x+6) \). What is the minimum value of \( g(x) \)?
参考答案
顶点・对称轴与最值挑战
\( f(x)=2x^2+28x+99 \)
The function g is defined by \( g(x)=f(x+6) \). What is the minimum value of \( g(x) \)?
The function g is defined by \( g(x)=f(x+6) \). What is the minimum value of \( g(x) \)?
参考答案
顶点・对称轴与最值挑战
\( f(x)=(x-6)(x+11) \)
The function \( f \) is defined by the given equation. For what value of x does \( f(x) \) reach its minimum?
The function \( f \) is defined by the given equation. For what value of x does \( f(x) \) reach its minimum?
参考答案
顶点・对称轴与最值挑战
Function \( f \) is a quadratic function. The graph of \( y=f\left(x\right) \) in the xy-plane has a vertex at \( \left(4,-6\right) \), contains the point \( \left(3,-8\right) \), and has a y-intercept at \( \left(0,\ a\right) \) . The graph of \( y=5\ ·\ f\left(x\right) \) has a y-intercept at \( \left(0,\ b\right) \). What is the positive difference between a and b?
原库参考解析
顶点・对称轴与最值进阶
\( f\left(x\right)=5x^2+90x+406 \)
The function g is defined by \( g\left(x\right)=f\left(x+8\right) \) . What is the minimum value of g(x)?
原库参考解析
顶点・对称轴与最值挑战
The graph of the equation \( y=5(x-n)^2 \), where n is an integer constant, is shown. What is the value of n?
原库参考解析
顶点・对称轴与最值挑战
\( f(x)=2x^2-17x+33 \)
The given equation defines the function \( f \). For what value of x does \( f(x) \) reach its minimum?
原库参考解析
