To find the zeros of the function \( i(x) = \frac{12x^3}{5} - \frac{27x}{5} \), we need to set the function equal to zero and solve for \( x \).
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Set the function equal to zero:
\[ \frac{12x^3}{5} - \frac{27x}{5} = 0 \]
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Multiply through by \( 5 \) to eliminate the fraction:
\[ 12x^3 - 27x = 0 \]
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Factor the equation:
\[ 3x(4x^2 - 9) = 0 \]
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This gives us two parts to solve for \( x \):
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Set \( 3x = 0 \): \[ x = 0 \]
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Set \( 4x^2 - 9 = 0 \): \[ 4x^2 = 9 \quad \Rightarrow \quad x^2 = \frac{9}{4} \quad \Rightarrow \quad x = \pm\frac{3}{2} \]
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So, the zeros of \( i(x) \) are:
- \( x = 0 \)
- \( x = \frac{3}{2} \)
- \( x = -\frac{3}{2} \)
Summary of the Zeros
The zeros of \( i(x) = \frac{12x^3}{5} - \frac{27x}{5} \) are \( x = 0 \), \( x = \frac{3}{2} \), and \( x = -\frac{3}{2} \).
Graphing the Zeros
To plot these zeros on a coordinate plane (Graph 2), follow these instructions:
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Draw the Axes: Start by drawing the x-axis and y-axis on a piece of graph paper.
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Label the Zeros: Place points on the x-axis at the following coordinates:
- \( (0, 0) \) - for \( x = 0 \)
- \( \left( \frac{3}{2}, 0 \right) \) - for \( x = \frac{3}{2} \). (This is approximately \( (1.5, 0) \))
- \( \left( -\frac{3}{2}, 0 \right) \) - for \( x = -\frac{3}{2} \). (This is approximately \( (-1.5, 0) \))
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Label Your Graph: Write "Graph 2" in a corner of your graph paper.
Visual Representation
You should see three points along the x-axis where \( i(x) = 0 \):
- Point at \( (0, 0) \)
- Point at \( (1.5, 0) \)
- Point at \( (-1.5, 0) \)
As you continue working on your portfolio, these points will form part of your complete graph for the function. If you need to sketch more details later, you can plot the entire function on the graph.
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