二次函数几种特殊形式的图象和性质
函数形式 | 顶点坐标 | 对称轴 | 最值 | 开口、单调性 |
$y = a x ^ {2}$ | $( 0,0 )$ | $y$轴 | $\left. \begin{array} {l} {a > 0 \text {时}} \\ {x = 0 \text {时,} y _ {\text {最小值}} = 0 ;} \\ {a < 0 \text {时}} \\ {x = 0 \text {时},y _ {\text {最大值}} = 0} \end{array} \right.$ | $a>0$时,抛物线开口向; $x$在对称轴侧时,$y$随$x$的增大而增大; $x$在对称轴侧时,$y$随$x$的增大而减小; $a<0$时,抛物线开口向,$x$在对称轴侧时,$y$随$x$的增大而增大; $x$在对称轴侧时,$y$随$x$的增大而减小 |
$y = a x ^ {2} + k$ | $( 0,k )$ | $y$轴 | $\left. \begin{array} {l} {a > 0 \text {时}} \\ {x = 0 \text {时},y _ {\text {最小值}} = k ;} \\ {a < 0 \text {时}} \\ {x = 0 \text {时},y _ {\text {最大值}} = k} \end{array} \right.$ | |
$y = a ( x - h ) ^ {2}$ | $( h,0 )$ | $x=h$ | $\left. \begin{array} {l} {a > 0 \text {时,}} \\ {x = h \text {时},y _ {\text {最小值}} = 0 ;} \\ {a < 0 \text {时},} \\ {x = h \text {时},y _ {\text {最大偵}} = 0} \end{array} \right.$ | |
$y = a ( x - h ) ^ {2} + k$ | $( h,k )$ | $x=h$ | $\left. \begin{array} {l} {a > 0 \text {时}} \\ {x = h \text {时},y _ {\text {最小值}} = k ;} \\ {a < 0 \text {时},} \\ {x = h \text {时},y _ {\text {最大值}} = k} \end{array} \right.$ |