510 Review Questions and Answers

Draw QR Code In NoneUsing Barcode printer for Software Control to generate, create QR image in Software applications.

Quick Response Code Decoder In NoneUsing Barcode recognizer for Software Control to read, scan read, scan image in Software applications.

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

QR Code Creator In C#.NETUsing Barcode drawer for VS .NET Control to generate, create Quick Response Code image in Visual Studio .NET applications.

QR Code ISO/IEC18004 Encoder In .NETUsing Barcode creator for ASP.NET Control to generate, create QR Code JIS X 0510 image in ASP.NET applications.

24

QR-Code Creator In .NET FrameworkUsing Barcode creator for Visual Studio .NET Control to generate, create QR-Code image in .NET applications.

QR Code 2d Barcode Maker In VB.NETUsing Barcode generation for .NET Control to generate, create Quick Response Code image in Visual Studio .NET applications.

Question 24-1

Making Bar Code In NoneUsing Barcode encoder for Software Control to generate, create barcode image in Software applications.

Draw Barcode In NoneUsing Barcode printer for Software Control to generate, create barcode image in Software applications.

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

Draw Code 128B In NoneUsing Barcode creator for Software Control to generate, create Code 128 image in Software applications.

UPCA Encoder In NoneUsing Barcode encoder for Software Control to generate, create UCC - 12 image in Software applications.

Answer 24-1

ECC200 Maker In NoneUsing Barcode creator for Software Control to generate, create Data Matrix 2d barcode image in Software applications.

EAN13 Maker In NoneUsing Barcode generation for Software Control to generate, create EAN-13 Supplement 5 image in Software applications.

The parabola opens straight upward if and only if a > 0 The parabola opens straight downward if and only if a < 0

Print UPC E In NoneUsing Barcode encoder for Software Control to generate, create UPC-E Supplement 2 image in Software applications.

Data Matrix ECC200 Creation In NoneUsing Barcode encoder for Microsoft Excel Control to generate, create Data Matrix image in Excel applications.

Question 24-2

Data Matrix Generation In JavaUsing Barcode printer for BIRT reports Control to generate, create DataMatrix image in Eclipse BIRT applications.

Barcode Encoder In .NET FrameworkUsing Barcode encoder for Reporting Service Control to generate, create bar code image in Reporting Service applications.

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

Code39 Creator In .NET FrameworkUsing Barcode generator for Reporting Service Control to generate, create USS Code 39 image in Reporting Service applications.

EAN13 Maker In JavaUsing Barcode creation for Java Control to generate, create EAN-13 Supplement 5 image in Java applications.

Answer 24-2

Printing Bar Code In NoneUsing Barcode creator for Font Control to generate, create bar code image in Font applications.

Read European Article Number 13 In NoneUsing Barcode reader for Software Control to read, scan read, scan image in Software applications.

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 downward 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

Part Three 511 Question 24-4

Imagine another quadratic function in which x is the independent variable and y is the dependent variable If this function has a single real zero with multiplicity 2 where x = p, what is xv, the x-value of the vertex point on its graph

Answer 24-4

When a quadratic function has only one real zero, the parabola is tangent to the x axis at the vertex point That s also the x-value of the real zero Therefore, xv = p

Question 24-5

Suppose we come across the following quadratic function in binomial factor form, where x is the independent variable and y is the dependent variable: y = (x + 2)(x 4) Does the parabola representing this function in Cartesian coordinates open upward or downward

Answer 24-5

To determine this, we must get the right side of the equation in polynomial standard form by multiplying the binomials When we do that, we get y = x 2 2x 8 Because the coefficient of x 2 is positive, the parabola opens upward

Question 24-6

What are the real zeros of the function stated in Question 24-5 What are the coordinates (xv, yv) of the vertex point in its graph Is the vertex an absolute maximum or an absolute minimum

Answer 24-6

The zeros can be seen by looking at the original form of the function The right side of that equation is a product of binomials If we set it equal to 0, getting a quadratic equation in x, we have (x + 2)(x 4) = 0 The zeros of the function are the same as the roots of this quadratic Without doing any algebra or arithmetic, we can see that these roots are x = 2 or x = 4 To find the vertex point, let s remember the general polynomial standard form for a quadratic function: y = ax 2 + bx + c The x-coordinate of the vertex point, xv, can be found by the formula xv = b /2a