The given function is:
$f(x) = (x + 3)(x - 1)$

Expanding the Expression

Using the distributive property (FOIL method): $f(x) = x^2 - x + 3x - 3$

Simplifying: $f(x) = x^2 + 2x - 3$

Key Features of the Function

1. Degree: The highest exponent of $x$ is 2, so this is a quadratic function.

2. Roots (x-intercepts): Setting $f(x) = 0$, $(x + 3)(x - 1) = 0$ Solving for $x$: $x + 3 = 0 \quad \Rightarrow \quad x = -3$ $x - 1 = 0 \quad \Rightarrow \quad x = 1$ So, the roots are $x = -3$ and $x = 1$.

3. Vertex: The vertex of a quadratic function in standard form $ax^2 + bx + c$ is given by: $x = \frac{-b}{2a}$ Here, $a = 1$ and $b = 2$, so: $x = \frac{-2}{2(1)} = -1$ Substituting $x = -1$ into $f(x)$: $f(-1) = (-1)^2 + 2(-1) - 3 = 1 - 2 - 3 = -4$ So, the vertex is $(-1, -4)$.

4. Y-Intercept: Setting $x = 0$, $f(0) = (0+3)(0-1) = 3(-1) = -3$ So, the y-intercept is $(0, -3)$.

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Related Questions

1. How do you find the axis of symmetry for a quadratic function?
2. What are the different methods to factorize a quadratic equation?
3. How do you determine whether a quadratic function opens upwards or downwards?
4. What are the applications of quadratic functions in real life?
5. How can you convert a quadratic function from factored form to vertex form?

Tip:

The discriminant $b^2 - 4ac$ determines the number and type of roots for a quadratic equation.