Algebra Know-It-ALL

510 Review Questions and Answers

Here, the coefficient of x^2 is real, the coefficient of x is complex, and the stand-alone constant

is complex. The complete polynomial quadratic is

x^2 + (− 3 −j2)x+ (1 +j3)= 0

Chapter 24

Question 24-1

Consider the general form of a quadratic function where x is the independent variable, y is the

dependent variable, and a,b, and c are real numbers with a≠ 0:

y=ax^2 +bx+c

The graph of this function in Cartesian coordinates is always a parabola that opens either

straight upward or straight downward. How can we tell which way the parabola opens by

simply looking at a specific function of this type?

Answer 24-1

The parabola opens straight upward if and only if a > 0. The parabola opens straight down-

ward if and only if a < 0.

Question 24-2

Suppose we see a quadratic function written as shown in Question 24-1, with specific num-

bers in place of a,b, and c. We plot several points (x,y) on the Cartesian plane by plugging in

various values of x and calculating the results for y. How can we determine how many real zeros

the function has, assuming we plot enough points to get a “clear picture” of the parabola?

Answer 24-2

The quadratic function has two different real zeros if and only if the parabola crosses the x axis

twice. The function has one real zero with multiplicity 2 if and only if the parabola is tangent to

(“brushes up against”) the x axis at the absolute maximum point or the absolute minimum point.

The function has no real zeros if and only if the parabola doesn’t intersect the x axis at all.

Question 24-3

Parabolas that open upward always have an absolute minimum. Parabolas that open down-

ward always have an absolute maximum. Imagine a quadratic function in which x is the

independent variable and y is the dependent variable. Its graph is a parabola. If the function

has two real zeros where x=p and x=q, what is the x-value of the absolute maximum or

minimum (that is, the vertex point) of the parabola? Let’s call it xv in this example.

Answer 24-3

The value xv is the average of the two zeros. That’s also known as the arithmetic mean, and is

equal to the sum of the values divided by 2:

xv= (p+q) / 2