College Algebra Final Exam Review – Step-by-Step Guidance

Q24. For the quadratic function $f(x) = -x^2 - 2x - 5$ :

Background

Topic: Quadratic Functions (Vertex Form)

This question tests your understanding of how to find the vertex of a parabola and how to rewrite a quadratic function in vertex form. The vertex form of a quadratic function is useful for identifying the vertex and understanding transformations.

Key Terms and Formulas:

• Standard form of a quadratic: $f(x) = ax^2 + bx + c$

• Vertex form: $f(x) = a(x-h)^2 + k$, where $(h, k)$ is the vertex

• Vertex formula: $h = -\frac{b}{2a}$, $k = f(h)$

Step-by-Step Guidance

1. Identify the coefficients: $a = -1$, $b = -2$, $c = -5$.

2. Find the x-coordinate of the vertex using $h = -\frac{b}{2a}$.

3. Substitute the value of $h$ back into $f(x)$ to find the y-coordinate $k$ of the vertex: $k = f(h)$.

4. Rewrite the function in vertex form by completing the square or using the vertex $(h, k)$ found above.

Try solving on your own before revealing the answer!

Final Answer:

a) Vertex: $(-1, -4)$

b) Vertex form: $f(x) = -(x + 1)^2 - 4$

The vertex is found using the formula $h = -\frac{b}{2a}$, and the function is rewritten by completing the square.