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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
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Question 24-1
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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
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Answer 24-1
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The parabola opens straight upward if and only if a > 0 The parabola opens straight downward if and only if a < 0
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Question 24-2
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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
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Answer 24-2
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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
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