Question

1. what is the graph of $y = x - 3$ (shown below) translated up 2 units? (there are four graphs labeled a, b, c, d with coordinate planes and parabolas)

Explanation:

Answer:

To determine the graph of \( y = x^2 - 3 \) translated up 2 units, we first recall the transformation rule for vertical shifts: adding a constant \( k \) to the function \( y = f(x) \) shifts the graph up by \( k \) units if \( k > 0 \).

Step 1: Find the new equation after translation

The original function is \( y = x^2 - 3 \). Translating it up 2 units means we add 2 to the entire function:

$$ y = (x^2 - 3) + 2 $$

Simplifying this, we get:

$$ y = x^2 - 1 $$

Step 2: Analyze the vertex of the parabola

The original function \( y = x^2 - 3 \) is a parabola with vertex at \( (0, -3) \) (since the standard form of a parabola is \( y = ax^2 + bx + c \), and for \( y = x^2 - 3 \), \( a = 1 \), \( b = 0 \), \( c = -3 \), so the vertex is at \( (-\frac{b}{2a}, f(-\frac{b}{2a})) = (0, -3) \)).

After translating up 2 units, the new vertex will be at \( (0, -3 + 2) = (0, -1) \).

Step 3: Match the graph with the new vertex

We now look at the options:

• Option a: The vertex appears to be at \( (0, -2) \) (incorrect, we need \( (0, -1) \)).
• Option b: The vertex is at \( (0, -1) \) (matches our calculation).
• Option c: The vertex is at \( (0, 0) \) (incorrect).
• Option d: The vertex is at \( (0, 0) \) (incorrect).

So the correct graph is option b.

Final Answer

b (the graph with vertex at (0, -1))