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
The function\( f\left(x\right)=\frac{1}{8}\left(x-6\right)^2+4 \) 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
Function f is a quadratic function. The graph of y = f (x) in the xy-plane has a vertex at (-5, -8), contains the point (-6, -10), and has a y-intercept at (0, a).The graph of y = 6 · 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・对称轴与最值挑战
The function f is a quadratic function. In the xy-plane, the graph of y = f (x) has a vertex at (1, 6) and passes through the points (2, 34) and (-1, 118). What is the value of f (-2) - f (0)?
原库参考解析
顶点・对称轴与最值挑战
The function \( f\left(x\right)=\frac{7}{8}\left(x-5\right)^2+7 \) predicts the speed \( f(x) \), in meters per second, of a bee flying \( x \) seconds after it takes off, where \( x \) > 0. 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・对称轴与最值进阶
\( f\left(x\right)=\left(x-4\right)^2+6 \)
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
\( f\left(x\right)=\left(x-1\right)^2+7 \)
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 \( y=f\left(x\right) \)has a vertex at \( \left(1,\ 4\right) \) and passes through the points \( \left(2,\ 27\right) \) and \( \left(-1,\ 96\right) \). What is the value of \( f\left(-2\right)-f\left(0\right) \)?
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\( y=f\left(x\right) \) has a vertex at \( \left(1,\ 6\right) \) and passes through the points \( \left(2,\ 40\right) \) and \( \left(-1,\ 142\right) \). What is the value of \( f\left(-2\right)-f\left(0\right) \) ?
原库参考解析
顶点・对称轴与最值进阶
The function \( f \) is a quadratic function. In the xy-plane, the graph of\( y=f\left(x\right) \) has a vertex at \( \left(1,\ 7\right) \) and passes through the points \( \left(2,\ 53\right) \) and \( \left(-1,\ 191\right) \). What is the value of \( f\left(-2\right)-f\left(0\right) \)?
原库参考解析
顶点・对称轴与最值进阶
\( y=x^2-24x+36 \)
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
\( f\left(x\right)=3x^2+48x+193 \)
The function \( g \) is defined by \( g\left(x\right)=f\left(x+7\right) \). What is the minimum value of \( g\left(x\right) \) ?
原库参考解析
顶点・对称轴与最值挑战
The function \( f(x)=\frac{1}{7}(x−6)^{2}+4 \) 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
