Find the minimum value of y = x2 - 2x - 3

1. Analyze the question

We are asked to find the minimum value of the quadratic function $y = x^2 - 2x - 3$. Since the coefficient of $x^2$ is positive ($+1$), the parabola opens upwards, meaning it has a minimum point at its vertex.

2. Explain the concept

For a quadratic equation in the form $y = ax^2 + bx + c$, the $x$-coordinate of the vertex (where the minimum or maximum occurs) is given by the formula:

Once you find the value of $x$, you substitute it back into the original equation to find the corresponding $y$-value, which is the minimum value of the function.

Alternatively, you can use the method of completing the square or differentiation.

3. Show the solution

Method 1: Using the vertex formula Given: $y = x^2 - 2x - 3$ Here, $a = 1$, $b = -2$, and $c = -3$.

• Step 1: Find the $x$-coordinate of the vertex. $$x = -\frac{b}{2a} = -\frac{-2}{2(1)} = \frac{2}{2} = 1$$

• Step 2: Substitute $x = 1$ into the equation to find $y$. $$y = (1)^2 - 2(1) - 3$$ $$y = 1 - 2 - 3$$ $$y = -1 - 3$$ $$y = -4$$

Method 2: Using Differentiation

• Step 1: Find the derivative $\frac{dy}{dx}$ and set it to zero. $$\frac{dy}{dx} = 2x - 2$$ $$2x - 2 = 0 \implies 2x = 2 \implies x = 1$$
• Step 2: Substitute $x = 1$ into the original equation. $$y = (1)^2 - 2(1) - 3 = -4$$

4. Identify the answer

Comparing our result to the given options: A. 4 B. 1 C. -1 D. -4

The calculated minimum value is -4, which corresponds to option D.

5. Tip

When you see a quadratic equation and need the minimum or maximum value, remember that the "value" requested is the $y$-coordinate, not the $x$-coordinate. Many students stop after finding $x = 1$ and mistakenly pick option B. Always substitute your $x$ back into the original equation to get the final answer!