3.3 Discriminants (Part 1)

Consider the quadratic function [math]f(x)=mx^2+4x+3[/math] where [math]m[/math] is a real number. If [math]f(x)[/math] is always positive when [math]a<m<b[/math], what is the value of [math]a+b[/math]?

The figure shows the graph of the quadratic function [math]f(x)=ax^2+bx+c[/math]. Identify the following signs. Write “[math]+[/math]”, “[math]-[/math]”, “[math]0[/math]” or “unknown”
(a) [math]a[/math]

(b) [math]b[/math]

(c) [math]c[/math]

(d) [math]b^2-4ac[/math]

(e) [math]a+b+c[/math]

(f) [math]a-b+c[/math]

(g) [math]9a+3b+c[/math]

(h) [math]4a+2b[/math]

Consider a quadratic function [math]f(x)=ax^2+bx+c[/math] where [math]a, b, [/math] and [math]c[/math] are constants. If [math]a<0[/math] and [math](3,15)[/math] is the vertex of [math]f(x)[/math], which of the following value(s) must be positive?

Consider a quadratic function [math]f(x)=x^2+2x+3[/math]. If line [math]\ell: y=x+k[/math] is a tangent line of [math]f(x)[/math] (i.e. [math]\ell[/math] touches [math]f(x)[/math]) where [math]k[/math] is a constant, what is the value of [math]k[/math]?

Let [math]f(x)=-x^2+kx+1[/math] be a quadratic function where [math]k[/math] is an integer. If the graph of [math]f(x)[/math] is below the horizontal line [math]y=3[/math], what is the maximum value of [math]k[/math]?

Let [math]f(x)[/math] and [math]g(x)[/math] be two quadratic functions with the same [math]x[/math]-intercept. Which statement(s) is/are true?

Find the constant [math]k[/math] such that the equation [math]x^2=x+k[/math] has only one real solution.

The figure shows the graph of the quadratic function [math]f(x)=ax^2+bx+c[/math]. Identify the following signs. Identify the following signs. Write “[math]+[/math]”, “[math]-[/math]”, “[math]0[/math]” or “unknown”
(a) [math]a[/math]

(b) [math]b[/math]

(c) [math]c[/math]

(d) [math]b^2-4ac[/math]

(e) [math]a+b+c[/math]

(f) [math]a-b+c[/math]

If [math]f(x)=ax^2+bx+3[/math] and has exactly one zero at [math]x=\frac{1}{2}[/math]. If [math]a\neq0[/math], find [math]ab[/math].

For what is the maximum integer [math]k[/math] would the graph of [math]y=x^2-2x+k[/math] cut the x-axis twice?

How many [math]x[/math]-intercepts does [math]f(x)=16x^2-8x+5[/math] have?

Which of the following equations has two complex, nonreal roots?

The graph of [math]f(x)=ax^2+bx+c[/math] is shown below.

(a) Find the value of [math]a[/math].

(b) The value of [math]b^2-4ac[/math] is: (write “[math]+[/math]”, “[math]-[/math]”, or “[math]0[/math]”)

(c) The roots of [math]f(x)[/math] are: (write “real” or “nonreal”)

(d) What is the maximum value of [math]f(x)[/math]?

(e) [math]f(6)=[/math]

How many [math]x[/math]-intercepts does [math]g(x)=-(x-30)^2+15[/math] have?

If [math]x^2+4x=3k[/math], what value of [math]k[/math] can make the equation has only one root?

For which of the following values of [math]a[/math] and [math]c[/math] does the equation [math]ax^2+3x+c=0[/math] have no real roots?

Given a quadratic function [math]f(x)=ax^2+bx+c[/math] with [math]a<0[/math] and two zeros at [math]x=-5[/math] and [math]x=3[/math], which of the following descriptions about [math]f(x)[/math] is correct?