To determine how the change in the equation affects the graph of the parabola, let's consider the original equation [tex]\( y = x^2 \)[/tex] and the new equation [tex]\( y = x^2 - 3 \)[/tex].
1. Start with the original function:
- The graph of [tex]\( y = x^2 \)[/tex] is a standard parabola that opens upwards with its vertex at the origin [tex]\((0, 0)\)[/tex].
2. Analyze the transformation:
- The new function is [tex]\( y = x^2 - 3 \)[/tex].
- Comparing this with the original function, we see that the term [tex]\(-3\)[/tex] is added to the original function.
3. Interpret the transformation term:
- Adding or subtracting a constant from the function [tex]\( y = x^2 \)[/tex] results in a vertical shift of the graph.
- In [tex]\( y = x^2 - 3 \)[/tex], the [tex]\(-3\)[/tex] term means that every value of [tex]\( y \)[/tex] in the original graph is decreased by 3.
4. Describe the effect on the graph:
- Subtracting 3 from each [tex]\( y \)[/tex]-value shifts the entire graph downward by 3 units. This means the vertex of the parabola moves from the origin [tex]\((0, 0)\)[/tex] to [tex]\((0, -3)\)[/tex].
5. Evaluate other options:
- The parabola does not become narrower or wider. These changes would involve multiplication factors affecting [tex]\( x^2 \)[/tex].
- The parabola also does not shift upward, as we are subtracting a constant, causing a downward shift.
Hence, the effect of changing the equation from [tex]\( y = x^2 \)[/tex] to [tex]\( y = x^2 - 3 \)[/tex] is that the parabola shifts 3 units down. Therefore, the correct answer is:
- The parabola shifts 3 units down.
