3.1 Quadratic functions (Part 1)

1. Question

Which of the following quadratic functions has/have maximum?

[math]y=(x-2)(x-3)[/math]

[math]y=3(6-x)^2-17[/math]

[math]y=9-x^2[/math]

[math]y=x(x-1)[/math]

[math]y=-6x^2+9x+17[/math]

2. Question

Let [math]f(x)=3(x+2)^2-6[/math] be a quadratic function. If [math]f(6)=m[/math], which of the following numbers is also [math]m[/math]?

[math]f(-2)[/math]

[math]f(-6)[/math]

[math]f(-8)[/math]

[math]f(-10)[/math]

None of the above.

3. Question

Let [math]f(x)[/math] be a quadratic function with [math]y[/math]-intercept [math]6[/math]. If [math]1[/math] and [math]5[/math] are the [math]x[/math]-intercepts of [math]f(x)[/math], what is the equation of [math]f(x)[/math]?

[math]f(x)=1.2(x-1)(x-5)[/math]

[math]f(x)=(x-1)(x-5)+1[/math]

[math]f(x)=1.2(x+1)(x+5)[/math]

[math]f(x)=(x+1)(x+5)+1[/math]

4. Question

Let [math]f(x)=a(x+5)(x-1)[/math] where [math]a[/math] is a negative number. If the range of [math]f(x)[/math] is [math](-\infty,18][/math], what is the value of [math]a[/math]?

5. Question

The figure below shows the graph of quadratic function [math]f(x)=ax^2+bx+c[/math] where [math]a, b[/math], and [math]c[/math] are constants. The value of [math]a+b+c[/math] is

Positive

Zero

Negative

Unknown

6. Question

Let [math]f(x)=a(x-p)(q-x)[/math] be a quadratic function where [math]a, p, q\in\mathbb{R}[/math] and [math]a\neq0[/math]. Which statement(s) is/are true?

[math]p[/math] and [math]q[/math] are the [math]x[/math]-intercepts of [math]f(x)[/math].

If [math]a>0[/math], then [math]f(x)[/math] has a minimum.

The vertical line [math]x=\dfrac{p+q}{2}[/math] is the axis of symmetry of [math]f(x)[/math].

[math]f(x)[/math] is always positive.

None of the above.

7. Question

Let [math]f(x)[/math] be a quadratic function. Suppose the [math]y[/math]-intercept of [math]f(x)[/math] is 0 and [math]x=-8[/math] is the axis of symmetry. Find the nonzero [math]x[/math]-intercepts of [math]f(x)[/math].

8. Question

Let [math]f(x)[/math] be a quadratic function with vertex [math](1,-2)[/math]. Suppose both the [math]x[/math]-intercepts of [math]f(x)[/math] are positive. Which quadrant does the function [math]f(x)[/math] fail to pass?

I

II

III

IV

9. Question

If quadratic function [math]f(x)[/math] has only one [math]x[/math]-intercept [math]-1[/math], which of the following equations could be [math]f(x)[/math]?

[math]f(x)=x^2-1[/math]

[math]f(x)=x^2+1[/math]

[math]f(x)=(x-1)^2[/math]

[math]f(x)=(x+1)^2[/math]

[math]f(x)=-x^2[/math]

10. Question

Let [math]f(x)[/math] be a quadratic function. Suppose [math]f(-5)=f(5)=4[/math]. Which value cannot be the [math]y[/math]-intercept?

[math]5[/math]

[math]-5[/math]

[math]4[/math]

The [math]y[/math]-intercept can be any number.

11. Question

(Calculator) Consider a parabolic cup. We pour some water into the cup. When the depth of the water is 8cm, the diameter of the water surface is 6cm. Suppose we keep pouring water into the cup until the depth of the water reaches 10cm. If the diameter of the water surface is [math]a\sqrt{b}[/math] where [math]a[/math] and [math]b[/math] are prime numbers, what are [math]a[/math] and [math]b[/math]?

[math]a=[/math] , [math]b=[/math]

12. Question

The figure below shows the graph of [math]f(x)=ax^2+bx+c[/math] where [math]a, b[/math], and [math]c[/math] are constants. What is the value of [math]a[/math]?

13. Question

Which of the following descriptions is incorrect about [math]g(x)=-3x^2+4[/math]?

The function has a minimum.

The axis of symmetry of [math]g(x)[/math] is [math]x=0[/math].

[math]g(x)[/math] is [math]y=-3x^2[/math] vertically shifted up by 4 units.

The discriminant of [math]g(x)[/math] is positive.

14. Question

Given [math]f(x)=-5(x-4)^2+1[/math], which of the following is the function after vertically shifting [math]f(x)[/math] down by 10 units and horizontally shifting [math]f(x)[/math] to the left by 2 units?