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・对称轴与最值挑战
Function \( f \) is a quadratic function. The graph of \( y=f(x) \) in the xy-plane has a vertex at (4, -6), contains the point (3, -8) , and has a y-intercept at (0, a) . The graph of \( y=5·f(x) \) has a y-intercept at (0, b). 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
At what value of x does the graph of \( y=x^2+18x-23 \) reach its minimum in the xy-plane?
原库参考解析
顶点・对称轴与最值进阶
The function \( f(x)=\frac{1}{9}(x-6)^2+3 \) gives a toy car’s height above the ground \( f(x) \), 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 \( y=f(x) \) in the xy-plane?
原库参考解析
顶点・对称轴与最值挑战
g(x)=(36-2x)(10+2x)
The function g is defined by the given equation. For what value of x does g(x) reach its maximum?
原库参考解析
顶点・对称轴与最值进阶
\( f\left(x\right)=x^2-12x+30 \)
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 graph of the quadratic function \( f \) is shown, where \( y=f(x) \). What is the vertex of the graph?
原库参考解析
顶点・对称轴与最值进阶
\( f(x)=10(x^2+47) \)
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
\( f(x)=4x^2+16x+22 \)
The function \( f \) is defined by the given equation. The function h is defined by \( h(x)=f(x)+9 \). What is the minimum value of \( h(x) \)?
原库参考解析
顶点・对称轴与最值进阶
\( f(x)=3x^2+12x+21 \)
The function \( f \) is defined by the given equation. The function h is defined by \( h(x)=f(x)+6 \). What is the minimum value of \( h(x) \)?
原库参考解析
顶点・对称轴与最值挑战
\( f(x)=(2x-15)(2x+7) \)
What is the minimum value of the given function?
原库参考解析
顶点・对称轴与最值挑战
The function \( f \) is a quadratic function. In the xy-plane, the graph of \( y=f(x) \) has a vertex at (1, 9) and passes through the points (2, 52) and (-1, 181). What is the value of \( f(−2)−f(0) \)?
原库参考解析
顶点・对称轴与最值进阶
\( f(x)=10\left(x^2+33\right) \)
What is the minimum value of the given function?
参考答案
