The graphs of the equations [tex]\(y = x^2\)[/tex] and [tex]\(y = 15 + x\)[/tex] have distinct characteristics and shapes, and they illustrate different mathematical properties and relationships between [tex]\(x\)[/tex] and [tex]\(y\)[/tex]. Let's explore these differences in detail:
1. Shape:
- The graph of [tex]\(y = x^2\)[/tex] is a parabola that opens upwards. It is U-shaped and is symmetric with respect to the y-axis.
- The graph of [tex]\(y = 15 + x\)[/tex] is a straight line with a slope of 1 and a y-intercept of 15. It is a linear function and thus forms a line.
2. Vertex and Intercept:
- The vertex of the parabola [tex]\(y = x^2\)[/tex] is at the origin (0, 0), which is the point where the parabola changes direction.
- The line [tex]\(y = 15 + x\)[/tex] has a y-intercept at (0, 15), which is the point where the line crosses the y-axis.
3. Slope:
- For [tex]\(y = x^2\)[/tex], the rate of change of [tex]\(y\)[/tex] with respect to [tex]\(x\)[/tex] is not constant. As [tex]\(x\)[/tex] increases or decreases, [tex]\(y\)[/tex] changes at an accelerating rate. The slope of the tangent to the parabola at any point increases as you move away from the origin.
- For [tex]\(y = 15 + x\)[/tex], the slope is constant and equal to 1. This means that for every unit increase in [tex]\(x\)[/tex], [tex]\(y\)[/tex] increases by exactly 1.
4. Growth Rate:
- In [tex]\(y = x^2\)[/tex], the value of [tex]\(y\)[/tex] grows quadratically as [tex]\(x\)[/tex] moves away from the origin. For example, when [tex]\(x = 2\)[/tex], [tex]\(y = 4\)[/tex]; when [tex]\(x = 3\)[/tex], [tex]\(y = 9\)[/tex].
- In [tex]\(y = 15 + x\)[/tex], the value of [tex]\(y\)[/tex] increases linearly. For example, when [tex]\(x = 2\)[/tex], [tex]\(y = 17\)[/tex]; when [tex]\(x = 3\)[/tex], [tex]\(y = 18\)[/tex].
5. Symmetry:
- The parabola [tex]\(y = x^2\)[/tex] is symmetric about the y-axis, meaning that if you fold the graph along the y-axis, both halves will match.
- The line [tex]\(y = 15 + x\)[/tex] does not have such symmetry. It is an increasing function that continuously rises from left to right.
6. Nature of Solutions:
- The equation [tex]\(y = x^2\)[/tex] has only one solution for [tex]\(x\)[/tex] when [tex]\(y\)[/tex] is positive or zero but has no real solution for negative values of [tex]\(y\)[/tex].
- The equation [tex]\(y = 15 + x\)[/tex] has one unique solution for [tex]\(x\)[/tex] for any real [tex]\(y\)[/tex].
In summary, while [tex]\(y = x^2\)[/tex] describes a quadratic relationship resulting in a parabolic graph, [tex]\(y = 15 + x\)[/tex] describes a linear relationship resulting in a straight line. These fundamental differences shape how each graph represents the relationship between [tex]\(x\)[/tex] and [tex]\(y\)[/tex].
