Here is the step-by-step solution to complete your homework.
Step 1: Complete the Table of Results
First, we need to calculate the value of $y$ for each $x$ by cubing the number (multiplying it by itself three times). The equation is $y = x^3$.
*
When $x = -2$:
$$(-2) \times (-2) \times (-2) = 4 \times (-2) = -8$$
*
When $x = -1$:
$$(-1) \times (-1) \times (-1) = 1 \times (-1) = -1$$
*
When $x = 0$:
$$0 \times 0 \times 0 = 0$$
*
When $x = 1$:
$$1 \times 1 \times 1 = 1$$
*
When $x = 2$:
$$2 \times 2 \times 2 = 8$$
The completed table looks like this:
| x | -2 | -1 | 0 | 1 | 2 |
| :--- | :---: | :---: | :---: | :---: | :---: |
|
$y = x^3$ |
-8 |
-1 |
0 |
1 |
8 |
*(Note: To finish the first part of the assignment, you would plot these points $(-2, -8), (-1, -1), (0, 0), (1, 1), (2, 8)$ on the graph paper and draw a smooth curve through them.)*
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Step 2: Solve the Equations using the Graph
Now we use the graph (or the logic of the cubic function) to find the answers. Since we don't have the drawn line in front of us, we can solve these algebraically or estimate based on the values we just calculated.
a) $5 = x^3$
We are looking for an $x$ value where the height ($y$) is $5$.
* From our table, we know that when $x=1$, $y=1$.
* When $x=2$, $y=8$.
* So, the answer must be between $1$ and $2$. Since $5$ is roughly in the middle of $1$ and $8$, the $x$ value will be around $1.7$.
* *Calculation check:* $\sqrt[3]{5} \approx 1.71$
b) $-6 = x^3$
We are looking for an $x$ value where the height ($y$) is $-6$.
* From our table, when $x=-1$, $y=-1$.
* When $x=-2$, $y=-8$.
* The answer is between $-1$ and $-2$. Since $-6$ is closer to $-8$ than it is to $-1$, the answer will be closer to $-2$. It is approximately $-1.8$.
* *Calculation check:* $\sqrt[3]{-6} \approx -1.82$
c) $y = 1.5^3$
This asks us to calculate the value directly rather than reading backwards from the graph. We need to cube $1.5$.
* First, multiply $1.5 \times 1.5 = 2.25$
* Next, multiply that result by $1.5$ again: $2.25 \times 1.5$
* $2.25 \times 1 = 2.25$
* $2.25 \times 0.5 = 1.125$
* $2.25 + 1.125 = 3.375$
Final Answer:
Table Values:
x: -2, y: -8
x: -1, y: -1
x: 0, y: 0
x: 1, y: 1
x: 2, y: 8
Equation Solutions:
a) $x \approx 1.7$
b) $x \approx -1.8$
c) $y = 3.375$
Parent Tip: Review the logic above to help your child master the concept of graphing cubic functions worksheet.
