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Chapter 8: Problem 20

The factored form of a quadratic function is \(y=-2(x-(3+i))(x-(3-i))\). Answer the following. a. Will the graph open up or down? Explain. b. What are the zeros of the quadratic function? c. Does the graph cross the \(x\) -axis? Explain. d. Write the quadratic in standard form. (Hint: Multiply out; see Exercise \(18 .\) ) e. Verify your answer in part (b) by using the quadratic formula and your answer for part (d).

Short Answer

Expert verified
a. The graph opens downwards. b. Zeros are \(3+i\) and \(3-i\). c. The graph does not cross the \(x\)-axis. d. Quadratic in standard form: \(y = -2x^2 + 12x - 20\). e. Zeros verified as \(3+i\) and \(3-i\).

Step by step solution

01

Determine If the Graph Opens Up or Down

Look at the coefficient of the quadratic term in factored form. The function given is \(y = -2(x-(3+i))(x-(3-i))\). The coefficient of the quadratic term is \(-2\). Since this coefficient is negative, the graph opens downwards.
02

Find the Zeros of the Quadratic Function

To find the zeros of the function, set \(y\) to zero and solve for \(x\): \(-2(x-(3+i))(x-(3-i)) = 0\). The solutions are \(x = 3+i\) and \(x = 3-i\). These are the zeros.
03

Determine if the Graph Crosses the x-axis

A graph of a quadratic function crosses the \(x\)-axis where the zeros are real numbers. In this case, the zeros are \(3+i\) and \(3-i\), which are complex numbers. Therefore, the graph does not cross the \(x\)-axis.
04

Write the Quadratic in Standard Form

To write the quadratic function in standard form, multiply out the factors: \(y = -2(x-(3+i))(x-(3-i))\). This can be simplified as follows: \(y = -2[(x-3-i)(x-3+i)]\). Using the difference of squares formula: \(y = -2[(x-3)^2 - i^2]\). Since \(i^2 = -1\), substitute to get \(y = -2[(x-3)^2 + 1]\). Expanding \((x-3)^2\), we get \((x-3)^2 = x^2 - 6x + 9\). Therefore, \(y = -2(x^2 - 6x + 10)\). Multiplying by \(-2\) gives: \(y = -2x^2 + 12x - 20\). Thus, the quadratic function in standard form is \(y = -2x^2 + 12x - 20\).
05

Verify Zeros Using the Quadratic Formula

To verify the zeros using the quadratic formula where the quadratic function is \(y = -2x^2 + 12x - 20\), use the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Here, \(a = -2\), \(b = 12\), and \(c = -20\). First, calculate the discriminant: \(b^2 - 4ac = 12^2 - 4(-2)(-20) = 144 - 160 = -16\). Next, using the quadratic formula, we get: \(x = \frac{-12 \pm \sqrt{-16}}{2(-2)} = \frac{-12 \pm 4i}{-4}\). Simplifying this: \(x = 3 \pm i\), which verifies the zeros as \(3+i\) and \(3-i\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

quadratic function
A quadratic function is a type of polynomial where the highest exponent of the variable is 2. The general form is written as: \ \[ y = ax^2 + bx + c \] Where:
  • \( a \) is the coefficient of the quadratic term \( x^2 \)
  • \( b \) is the coefficient of the linear term \( x \)
  • \( c \) is the constant term

Quadratic functions have a parabolic shape, meaning they look like a 'U' which can either go upwards or downwards. If \( a \) is positive, the parabola opens upwards; if \( a \) is negative, it opens downwards. A key feature of quadratic functions is their vertex, which is the highest or lowest point on the graph. This vertex is found using the formula: \ \[ x = -\frac{b}{2a} \] Understanding the direction and shape of a quadratic function is fundamental to analyzing its behavior and solving related problems.
complex zeros
Complex zeros of a quadratic function are solutions to the equation that are not real numbers. They include the imaginary unit \( i \), where \( i^2 = -1 \). Complex zeros come in conjugate pairs, such as \( 3+i \) and \( 3-i \).
To find complex zeros, set the quadratic function equal to zero and solve for the variable \( x \). For example, given the quadratic function in factored form: \ \[ y = -2(x-(3+i))(x-(3-i)) \] Set \( y \) to zero: \[ -2(x-(3+i))(x-(3-i)) = 0 \] The solutions are \( x = 3+i \) and \( x = 3-i \). These values are the complex zeros. Since complex zeros are not real, their corresponding points do not lie on the \( x \)-axis. This means the parabola described by the quadratic function will not cross the \( x \)-axis.
standard form of quadratic
A quadratic function can be written in different forms. One important form is the standard form, which is: \ \[ y = ax^2 + bx + c \] Example: starting with the factored form: \[ y = -2(x-(3+i))(x-(3-i)) \] We first expand the factors using the difference of squares formula: \[ (x-(3+i))(x-(3-i)) = (x-3-i)(x-3+i) = (x-3)^2 - i^2 \] Since \( i^2 = -1 \), substituting yields: \[ (x-3)^2 + 1 \] Next, expand \( (x-3)^2 \): \[ (x-3)^2 = x^2 - 6x + 9 \] Therefore, the factored form becomes: \[ y = -2(x^2 - 6x + 10) = -2x^2 + 12x - 20 \] Thus, the quadratic function in standard form is \( y = -2x^2 + 12x - 20 \). Writing the function in standard form is invaluable for identifying key characteristics such as the vertex and the direction in which the graph opens.

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Most popular questions from this chapter

Find the equation of the graph of a parabola that has the following properties: \(\bullet\) The \(x\) -intercepts of the graph are at (2,0) and \((3,0),\) and \(\bullet\) The parabola is the graph of \(y=x^{2}\) vertically stretched by a factor of 4 .

For each of the following quadratics with their respective vertices, calculate the distance from the vertex to the focal point. Then determine the coordinates of the focal point. a. \(f(x)=x^{2}-2 x-3\) with vertex at (1,-4) b. \(g(t)=2 t^{2}-16 t+24\) with vertex at (4,-8)

Write each of the following quadratic equations in function form (i.e., solve for \(y\) in terms of \(x\) ). Find the vertex and the \(y\) -intercept using any method. Finally, using these points, draw a rough sketch of the quadratic function. a. \(y+12=x(x+1)\) d. \(y-8 x=x^{2}+15\) b. \(2 x^{2}+6 x+14.4-2 y=0\) e. \(y+1=(x-2)(x+5)\) c. \(y+x^{2}-5 x=-6.25\) f. \(y+2 x(x-6)=20\)

Transform the function \(f(x)=3 x^{2}\) into a new function \(h(x)\) by shifting \(f(x)\) horizontally to the right four units, reflecting the result across the \(x\) -axis, and then shifting it up by five units. a. What is the equation for \(h(x) ?\) b. What is the vertex of \(h(x) ?\) c. What is the vertical intercept of \(h(x) ?\)

If a parabola is the graph of the equation \(y=a(x-4)^{2}-5\) : a. What are the coordinates of the vertex? Will the vertex change if \(a\) changes? b. What is the value of stretch factor \(a\) if the \(y\) -intercept is (0,3)\(?\) c. What is the value of stretch factor \(a\) if the graph goes through the point (1,-23)\(?\)
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