Question

Graph each function using the techniques of shifting, compressing, stretching, and/or reflecting. Start with the graph of the basic function (for example, \(y=x^{2}\) ) and show all the steps. Be sure to show at least three key points. Find the domain and the range of each function. $$ f(x)=2(x+1)^{2}-3 $$

Short answer

Domain: \[ (-\infty, \infty) \]. Range: \[ [-3, \infty) \].

Step-by-step solution

Identify the Basic Function

The basic function to start with is the quadratic function: \[ y = x^2 \]

Apply Horizontal Shift

Since the function is \[ f(x) = 2(x+1)^{2}-3 \], we identify \( (x+1) \) as a horizontal shift. This shifts the graph left by 1 unit. The new equation becomes: \[ y = (x+1)^2 \].

Apply Vertical Stretch

Next, notice the coefficient 2 in front of \( (x+1)^2 \). This coefficient vertically stretches the graph by a factor of 2. The updated function is \[ y = 2(x+1)^2 \].

Apply Vertical Shift

Finally, we apply the vertical shift by subtracting 3 from the previous function. This results in the final function: \[ y = 2(x+1)^2 -3 \]. This shifts the graph down by 3 units.

Plot Key Points

Identify key points on the transformed graph. Start with three key points from the basic function and apply the transformations.Original points (for \( y=x^2 \)) are: \((-1, 1)\), \( (0, 0), \) and \((1, 1)\).New transformed points are: (-2, -1), (-1, -3) and (0, 1). Plot these points on the graph.

Determine the Domain

The domain of this function is all real numbers, as there is no restriction on the values of \( x \). So, the domain is: \[ \text{Domain: } (-\infty, \infty) \].

Determine the Range

The range of the function is determined by the vertex of the parabola. Since the graph is opening upwards and the lowest point (vertex) is ( -1, -3 ), the range starts from -3 and goes to positive infinity. So, the range is: \[ \text{Range: } [-3, \infty) \].

Key concepts

horizontal shift

When we talk about horizontal shift, it means we are moving the graph left or right. For the function \(f(x) = 2(x+1)^{2}-3\), the term \(x+1\) indicates a shift to the left by 1 unit. This is because adding to the \(x\) term moves the graph in the opposite direction. Starting from the basic function \(y = x^2\), the new function after the horizontal shift is \(y = (x+1)^{2}\). Visually, every point on the graph of \(y = x^2\) moves one unit to the left. For example, the point \( (0, 0) \) on \( y = x^2 \) moves to \( (-1, 0)\) on \( y = (x+1)^{2} \).

vertical stretch

A vertical stretch changes the shape of the graph by making it taller. It scales the function values (the \(y\) values) by a factor. For our function \(f(x) = 2(x+1)^{2}-3\), the coefficient 2 in front of \( (x+1)^2 \) means the graph is stretched vertically by a factor of 2. This makes the parabola narrower compared to the standard \(y = x^2\). Every \(y\) coordinate from the basic function \( y=(x+1)^{2} \) gets multiplied by 2. For example, the point \( (-1, 1) \) on \( y = (x+1)^{2}\) changes to \( (-1, 2) \) on \( y = 2(x+1)^{2}\).

vertical shift

A vertical shift moves the graph up or down. This is controlled by adding or subtracting a value to the function. For \(f(x) = 2(x+1)^{2}-3 \), the \(-3\) indicates a vertical shift down by 3 units. This means the entire graph of \( y = 2(x+1)^{2} \) is moved 3 units down. So, every \(y\) coordinate is decreased by 3. For instance, the point \( (-1, 2) \) in \( y = 2(x+1)^{2} \) becomes \((-1, -1) \) in \( y = 2(x+1)^{2} -3\).

domain and range

The domain and range are crucial aspects of understanding a function.

**Domain**: The domain of a quadratic function is always all real numbers unless a restriction is indicated. For \( f(x) = 2(x+1)^{2} -3 \), there are no restrictions on \( x\). Thus, the domain is \[ (-\textbackslash infty, \textbackslash infty) \].

**Range**: The range of the function is determined by the vertex of the parabola. Here, the vertex is \((-1, -3)\). Since the parabola opens upwards, the \(y\) values start from -3 and go to infinity. Therefore, the range is \[ [-3, \textbackslash infty) \]. This tells us that the function's output will be \( y \) values starting from \( -3\) and above.