问题 解答题

已知抛物线y=8x2+10x+1

(1)试判断抛物线与x轴交点情况

(2)求此抛物线上一点A(-1,-1)关于对称轴的对称点B的坐标

(3)是否存在一次函数与抛物线只交于B点?若存在,求出符合条件的一次函数的解析式;若不存在,请说明理由.

答案

(1)令y=0,得8x2+10x+1=0,△=100-4×8>0;

因此抛物线与x轴有两个不同的交点.

(2)易知:抛物线的对称轴为x=-

5
8
,∴B(-
1
4
,-1)

(3)假设存在这样的一次函数,设一次函数的解析式为y=kx+b,已知直线过B点,则有:

-

1
4
k+b=-1,b=
k
4
-1,

∴y=kx+

k
4
-1.

依题意有:

y=8x2+10x+1
y=kx+
k
4
-1

则有8x2+10x+1=kx+

k
4
-1,

即8x2+(10-k)x+

8-k
4
=0;

由于两函数只有一个交点,

因此△=(10-k)2-8(8-k)=0,

即(k-6)2=0

∴k=6

∴一次函数的解析式为y=6x+

1
2

单项选择题 案例分析题
单项选择题

Weak dollar or no, $ 46,000-the price for a single year of undergraduate instruction amid the red brick of Harvard Yard-is (1) But nowadays cost is (2) barrier to entry at many of America’s best universities. Formidable financial-assistance policies have (3) fees or slashed them deeply for needy students. And last month Harvard announced a new plan designed to (4) the sticker-shock for undergraduates from middle and even upper-income families too.

Since then, other rich American universities have unveiled (5) initiatives. Yale, Harvard’s bitterest (6) , revealed its plans on January 14th. Students whose families make (7) than $60,000 a year will pay nothing at all. Families earning up to $ 200,000 a year will have to pay an average of 10% of their incomes. The university will (8) its financial- assistance budget by 43%, to over $ 80m.

Harvard will have a similar arrangement for families making up to $180,000. That makes the price of going to Harvard or Yale (9) to attending a state-run university for middle-and upper-income students. The universities will also not require any student to take out (10) to pay for their (11) , a policy introduced by Princeton in 2001 and by the University of Pennsylvania just after Harvard’s (12) . No applicant who gains admission, officials say, should feel (13) to go elsewhere because he or she can’t afford the fees.

None of that is quite as altruistic as it sounds. Harvard and Yale are, after all, now likely to lure more students away from previously (14) options, particularly state-run universities, (15) their already impressive admissions figures and reputations.

The schemes also provide a (16) for structuring university fees in which high prices for rich students help offset modest prices for poorer ones and families are less (17) on federal grants and government-backed loans.

Less wealthy private colleges whose fees are high will not be able to (18) Harvard or Yale easily. But America’s state-run universities, which have traditionally kept their fees low and stable, might well try a differentiated (19) scheme as they raise cash to compete academically with their private (20) . Indeed, the University of California system has already started to implement a sliding-fee scale.

5()

A.different

B.same

C.similar

D.encouraging