Question

Sketch the graph of the first function by plotting points if necessary. Then use transformation(s) to obtain the graph of the second function. \(y=x^{2}, \quad y=2 x^{2}-4 x+1\)

Short answer

To sketch the graph of \(y=x^2\), plot points such as (-1,1), (0,0), and (1,1) and form a basic upward-opening parabola. For the second function, \(y=2x^2-4x+1\), find its vertex \((1,-3)\) using the vertex formula. Then, transform the first graph by stretching it vertically by a factor of 2, translating left by one unit, and translating down by three units. This results in the second function's graph, \(y=2x^2-4x+1\), which is a steeper parabola with vertex (1,-3).

Step-by-step solution

Analyze the given functions

1. For the first function, \(y = x^2\), we can tell that it's a simple quadratic function (a parabola) with the vertex at the origin \((0,0)\) and opens upwards because the leading coefficient (1) is positive. 2. For the second function, \(y = 2x^2 - 4x + 1\), we see that it's also a quadratic function, and it will also have a parabolic shape with its vertex shifted from the origin. We will find the vertex later to guide us in the transformation process.

Sketch the graph of the first function

To sketch the graph of \(y=x^2\), we can plot a few points. Consider the points (-1,1), (0,0), and (1,1) on the graph. These points show a basic upward-opening parabola.

Find the vertex of the second function

To find the vertex of the second function, which is in the form \(y = ax^2+bx+c\), we can use the formula for the vertex, \((h,k)\), given by: \(h = \frac{-b}{2a} \) and \(k = c - \frac{b^2}{4a}\) For the given function, \(y=2x^2-4x+1\), we have \(a = 2\), \(b = -4\), and \(c = 1\). Plugging these values into the vertex formula gives: \(h = \frac{-(-4)}{2(2)} = \frac{4}{4} = 1 \) \(k = 1 - \frac{(-4)^2}{4(2)} = 1 - 4 = -3\) So, the vertex of the second function is \((1,-3)\).

Use transformations to obtain the graph of the second function

Now that we have the vertex of the second function, we can use transformations to obtain its graph from the sketch of the first function, \(y=x^2\). Notice that the second function has a leading coefficient of 2, which will make the parabola narrower and steeper. Therefore, we can apply the following transformations: 1. Stretch the parabola vertically by a factor of 2: \(y = 2x^2\) 2. Translate the parabola left by one unit: \(y = 2(x - 1)^2\) 3. Translate the parabola down by three units: \(y = 2(x - 1)^2 - 3\) At this point, you should have the second function's graph, \(y = 2x^2-4x+1\), which is a steeper parabola with vertex (1,-3). By following these steps and applying the transformations, you can sketch the graph of the given functions.