Question

Graph the following on the same coordinate system. (a) y = x2 (b) y = 3x2 (c) y = 1/3x2 (d) How does the coefficient of x2 affect the shape of the graph?

Accepted Answer

Identify the base function for all graphs, which is the quadratic function $$y = x^2$. This is a parabola opening upwards with its vertex at the origin (0$$,0)$$. For the function y$$ = $$3x^2$$, recognize that the coefficient 3 is greater than 1, which causes the parabola to become narrower (steeper) compared to $$y = x^2$. Plot points by substituting values of x$ and calculating y$ to see this effect. For the function y$$ = \frac{1}{3}$$x^2$$, note that the coefficient $$\frac{1}{3} is $$between 0 and 1, which makes the parabola wider (less steep) than $$y = x^2$. Again, plot points by substituting values of x$ to observe this change. Graph all three functions on the same coordinate system by plotting several points for each and drawing smooth parabolas through these points, ensuring the vertex is at the origin for all. Analyze how the coefficient of x^2$ affects the shape: coefficients greater than 1 make the parabola narrower, coefficients between 0 and 1 make it wider, and a negative coefficient would reflect it across the x-axis (not in this problem).

Graph the following on the same coordinate system.
(a) y = x²
(b) y = 3x²
(c) y = 1/3x²
(d) How does the coefficient of x² affect the shape of the graph?

Verified video answer for a similar problem:

Key Concepts

Quadratic Functions and Their Graphs

Effect of the Leading Coefficient on the Parabola

Graphing Multiple Functions on the Same Coordinate System