- Evaluate \(f(0)\) to find the \(y\)-intercept: \(f(0) = -12\).
- Set \(f(x) = 0\) and factor the cubic polynomial by grouping: \((x - 2)(x + 2)(x + 3) = 0\).
- Identify the roots of the equation to find the \(x\)-intercepts: \(x = -3, -2, 2\).
- Final answers: \(x\)-intercept(s): \(\boxed{{-3, -2, 2}}\), \(y\)-intercept(s): \(\boxed{{-12}}\).
Explanation
- Finding the y-intercept
To find the \(y\)-intercept of the function \(f(x) = x^3 + 3x^2 - 4x - 12\), we evaluate the function at \(x = 0\). Substituting \(x = 0\) into the expression:
\[f(0) = 0^3 + 3(0)^2 - 4(0) - 12 = -12.\]
Therefore, the \(y\)-intercept is \(-12\).
- Finding the x-intercepts
To find the \(x\)-intercepts of the function \(f(x) = x^3 + 3x^2 - 4x - 12\), we set the function equal to zero and solve the equation:
\[x^3 + 3x^2 - 4x - 12 = 0.\]
We can factor this cubic polynomial by grouping terms:
\[(x^3 + 3x^2) - (4x + 12) = 0.\]
Factoring out common terms from each group:
\[x^2(x + 3) - 4(x + 3) = 0.\]
Next, factor out the common binomial factor \((x + 3)\):
\[(x^2 - 4)(x + 3) = 0.\]
The expression \((x^2 - 4)\) is a difference of squares and can be factored as \((x - 2)(x + 2)\). Thus, the complete factored form of the function is:
\[(x - 2)(x + 2)(x + 3) = 0.\]
Setting each factor to zero gives us the \(x\)-intercepts:
\[x - 2 = 0 \implies x = 2\]
\[x + 2 = 0 \implies x = -2\]
\[x + 3 = 0 \implies x = -3\]
Thus, the \(x\)-intercepts are \(-3, -2, 2\).
- Conclusion and Final Results
We have found the \(y\)-intercept to be \(-12\) and the \(x\)-intercepts to be \(-3, -2,\) and \(2\). These points represent where the graph of the function intersects the axes. The results are summarized below:
\(x\)-intercept(s): \(-3, -2, 2\)
\(y\)-intercept(s): \(-12\)
Examples
Finding the intercepts of a function is crucial in various fields, such as engineering and economics. For example, if a function represents the path of a projectile, the \(x\)-intercepts tell you where the object hits the ground, while the \(y\)-intercept tells you the initial height of the launch. By calculating these points, analysts can predict outcomes and optimize designs for efficiency and safety.