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・对称轴与最值基础
\( f\left(x\right)=3x^2+36x+109 \)
The function g is defined by g(x)=f(x+7). What is the minimum value of g(x)?
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 \( f\left(x\right)=\frac{1}{9}\left(x-6\right)^2+3 \) 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
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 (9, -7), contains the point (8, -9), 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 quadratic function f has a factor of ( x +7.75) and a zero x = c, where c is a positive constant. The graph of y = f(x) in the xy-plane has its vertex at(-3.75,-d), where d is a positive constant. What is the value of c?
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 (1, 4) and passes through the points (2, 37) and (−1, 136). 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 graph of the quadratic function y=f(x) in the xy-plane intersects the x-axis when x=39 and when x=p, where p is a constant. The maximum value of y=f(x) occurs at the point (14, m), where m is a constant. What is the value of p?
原库参考解析
顶点・对称轴与最值挑战
The function g is a quadratic function. In the xy-plane, the graph of y=g(x) has a vertex at (-1, -4) and passes through the points (-2, -43) and (1, -160). What is the value of g(0)-g(2)?
原库参考解析
顶点・对称轴与最值进阶
\( f\left(x\right)=\left(x-65\right)\left(x-67\right) \)
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・对称轴与最值进阶
The town of Fillmore is planning its annual fireworks show. The equation \( y=-16\left(x-6.8\right)^2+740 \) gives the estimated height y, in feet, of a type of firework x seconds after it is launched into the air. Which of the following is the best interpretation of the vertex of the graph of the equation in the xy-plane?
原库参考解析
顶点・对称轴与最值进阶
The equation \( y=-4.9\left(x-8.8\right)^2+11,900 \) gives the estimated height above ground, y, in meters, of a plane, where x is the number of seconds since it started a parabolic maneuver. If this equation is graphed in the xy-plane, which of the following is the best interpretation of the vertex of the graph?
原库参考解析
顶点・对称轴与最值挑战
\( p\left(x\right)=2x^2+12x+19 \)
What is the minimum value of the given function?
原库参考解析
