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
\( f\left(x\right)=4x^2+64x+262 \)
The function g is defined by \( g\left(x\right)=f\left(x+5\right) \) . 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) = ax2 + 4x + c
In the given quadratic function, a and c are constants. The graph of y = f(x) 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 (-9)=f (3), which of the following must be true?
I. c < 0
II. a\( \ge \) 1
原库参考解析
顶点・对称轴与最值挑战
f(x) = 4x2 + 64x + 262
The function g is defined by g(x) = f (x+5) . For what value of x does g(x) reach its minimum?
原库参考解析
顶点・对称轴与最值挑战
y = x2 - 14x + 22
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
f(x) = (x -14)(x +19)
The function f is defined by the given equation. 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
The function \( f\left(x\right)=\frac{1}{9}\left(x-7\right)^2+3 \) gives a metal ball's height above the ground \( f(x) \), in inches, x seconds after it started moving on a track, where \( 0\le x\le10 \) . Which of the following is the best interpretation of the vertex of the graph of y = \( f(x) \) in the xy-plane?
原库参考解析
顶点・对称轴与最值挑战
When the quadratic function f is graphed in the xy-plane, where y = f(x) , its vertex is (-3 , 6). One of the x-intercepts of this graph is \( \left(-\frac{17}{4},\ 0\right) \) . What is the other x-intercept of the graph?
原库参考解析
顶点・对称轴与最值挑战
The function f is defined by f(x) =(x + 3)(x + 1). The graph of f in the xy-plane is a parabola. Which of the following intervals contains the x-coordinate of the vertex of the graph of f ?
原库参考解析
顶点・对称轴与最值挑战
f(x) = 4x2 - 50x + 126
The given equation defines the function 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(x) = (x + 7)2 +4
The function f is defined by the given equation. 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(x) = (x -10)(x +13)
The function f is defined by the given equation. 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
What is the minimum value of the function f defined by f(x) = (x -2)2 -4 ?
原库参考解析
