Question

Graph the function \(f\) by starting with the graph of \(y=x^{2}\) and using transformations. \(f(x)=2 x^{2}+4\)

Short answer

Stretch \( y = x^2 \) vertically by 2 and shift it upwards by 4 units.

Step-by-step solution

Identify the base function

The base function is given by the quadratic function, which is the graph of \( y = x^2 \). This is a parabola that opens upwards with its vertex at the origin (0,0).

Apply the vertical stretch

The function has a coefficient of 2 in front of the \( x^2 \) term. This means you need to vertically stretch the graph of \( y = x^2 \) by a factor of 2. This transformation changes the function to \( y = 2x^2 \).

Apply the vertical shift

Next, the function has a +4 added to the \( 2x^2 \). This means you need to shift the graph of \( y = 2x^2 \) upwards by 4 units. This transformation changes the function to \( y = 2x^2 + 4 \).

Sketch the transformed graph

Start by drawing the original graph \( y = x^2 \) as a reference. Then apply the vertical stretch to obtain the graph of \( y = 2x^2 \). Finally, shift this new graph upwards by 4 units to obtain the graph of \( y = 2x^2 + 4 \).

Key concepts

Vertical Stretch

A vertical stretch changes the shape of the graph by stretching or compressing it along the vertical axis. Specifically, multiplying the entire function by a coefficient greater than 1 causes the graph to stretch vertically. In our example, the function is given as \( f(x) = 2x^2 + 4 \). Here, multiplying by 2 stretches the graph of \( y = x^2 \) by a factor of 2. This means that each point on the graph of \( y = x^2 \), originally at \( (x, y) \), moves to \( (x, 2y) \). Thus, points like \( (1, 1) \) become \( (1, 2) \) and \( (2, 4) \) become \( (2, 8) \). This makes the parabola narrower but maintains the overall shape.

Vertical Shift

A vertical shift moves the entire graph of a function up or down without changing its shape. In our function \( f(x) = 2x^2 + 4 \), the +4 at the end moves the graph of \( y = 2x^2 \) up by 4 units. This means each point \( (x, y) \) on the graph of \( y = 2x^2 \) will move to \( (x, y + 4) \). For example, the vertex of \( y = 2x^2 \), which is at \( (0, 0) \) will move up to \( (0, 4) \). Similarly, the point \( (1, 2) \) on \( y = 2x^2 \) will move to \( (1, 6) \). This shifts the entire parabola upwards but maintains the same shape and orientation.

Quadratic Function

A quadratic function is a polynomial function of degree 2, typically in the form \( f(x) = ax^2 + bx + c \). Its graph is a parabola that can open upwards or downwards depending on the sign of the coefficient \( a \). For \( f(x) = x^2 \), the parabola opens upwards with a vertex at the origin \( (0, 0) \). In our example, \( f(x) = 2x^2 + 4 \), the function maintains its parabolic shape, opens upwards, and is influenced by the transformations: a vertical stretch by a factor of 2 and a vertical shift up by 4 units. These transformations are applied successively, modifying the original shape and position.

Graphing

Graphing is the process of drawing a function on a coordinate plane. Here’s how to graph \( f(x) = 2x^2 + 4 \):
- Start by drawing the base graph \( y = x^2 \), a standard parabola opening upwards with its vertex at \( (0, 0) \).
- Apply the vertical stretch: Transform this into \( y = 2x^2 \). Each y-coordinate is doubled, making the graph narrower.
- Apply the vertical shift: Move every point on \( y = 2x^2 \) up by 4 units to get \( y = 2x^2 + 4 \).
- The final graph maintains its parabolic shape but is now narrower and shifted up.