Problem 23 - Algebra and Trigonometry

Question

Use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola's axis of symmetry. Use the graph to determine the function's domain and range. $$ f(x)=2(x+2)^{2}-1 $$

Step-by-Step Solution

Answer

The vertex of the function is (-2,-1). The x-intercepts are \(-2 + \sqrt{1/2}\) and \(-2 - \sqrt{1/2}\). The equation of the axis of symmetry is \(x=-2\). The domain of the function is all real numbers, \(x ∈ (-∞, ∞)\), and the range is \(y ≥ -1\).

Step 1: Identify Vertex and Intercepts

The vertex form of a quadratic function is \( f(x)=a(x-h)^{2}+k \), where (h,k) is the vertex. Given \(f(x)=2(x+2)^{2}-1\), the vertex is (-2,-1). For x-intercept(s), set \(f(x) = 0\) and solve for \(x\). The function intercepts the x-axis when the \(y\) value is 0, which is when \(2(x+2)^{2}-1 = 0\). Solving this gives \(x = -2 \pm \sqrt{1/2}\). Therefore, the x-intercepts are \(-2 + \sqrt{1/2}\) and \(-2 - \sqrt{1/2}\).

Step 2: Sketch the Graph

First plot the vertex (-2,-1) on the graph. Then, plot the two points representing the x-intercepts (\(-2 + \sqrt{1/2}\), 0) and (\(-2 - \sqrt{1/2}\), 0). Connect these 3 points with a smooth curve to form your parabola. The graph should open upwards since the coefficient of \(x^2\) is positive.

Step 3: Determine Axis of Symmetry

The axis of symmetry of a parabola in the form \(f(x)=a(x-h)^{2}+k\) is \(x=h\). Therefore, the equation of the axis of symmetry is \(x=-2\).

Step 4: Determine Domain and Range

The domain of a quadratic function is all real numbers, as \(x\) can take any value. Therefore, the domain is \(x ∈ (-∞, ∞)\). The range of a quadratic function that opens upwards and has a vertex (h,k) is \(y ≥ k\). Therefore, the range is \(y ≥ -1\).

Key Concepts

Vertex Form of a Quadratic

Understanding the vertex form of a quadratic function is essential to sketching its graph. The vertex form is given by the equation \( f(x) = a(x-h)^2+k \), where \((h, k)\) represents the coordinates of the vertex of the parabola, and \(a\) determines the direction and width of the parabola. If \(a\) is positive, the parabola opens upward, and if \(a\) is negative, it opens downward.

The function \( f(x)=2(x+2)^{2}-1 \) is already in vertex form. Here, we can see the vertex \((-2, -1)\) can be easily identified. This point is the peak (if \(a\) is negative) or trough (if \(a\) is positive) of the parabola, and it plays a crucial role in drawing a precise graph. By finding the vertex, we can start plotting the parabola's shape and guide our sketch to include all other key features such as the axis of symmetry and intercepts.

Axis of Symmetry

The axis of symmetry in a quadratic function is a vertical line that divides the parabola into two mirror images. It passes through the vertex, meaning it has the same \(x\)-value as the vertex. In the vertex form \( f(x)=a(x-h)^{2}+k \), the axis of symmetry is simply \( x = h \).

For the function \( f(x)=2(x+2)^{2}-1 \), since \( h=-2 \), the axis of symmetry is the line \( x=-2 \). Identifying the axis of symmetry helps us create symmetrical plots on each side when sketching the parabola. Remember, any point on one side of the axis will have a corresponding point on the other side at the same distance from the axis.

Domain and Range of a Function

The domain of a function refers to all the possible values of \(x\) that can be plugged into the function, and the range refers to all the possible values of \(y\) that the function can output. For all quadratic functions, the domain is always all real numbers because you can select any value for \(x\) and get a corresponding \(y\) value.

In our specific function \( f(x)=2(x+2)^{2}-1 \), which is a parabola that opens upwards, the domain is \( x \in (-\infty, \infty) \). The range, however, is determined by the vertex which tells us the lowest point on the graph is at \(y = -1\). Since the parabola opens upwards, \(y\) can take on any value greater than or equal to -1. Therefore, the range is \( y \geq -1 \), which can be written as \([ -1, \infty )\).

X-Intercepts of a Parabola

The x-intercepts of a parabola are points where the graph intersects the x-axis. These occur where the function \(f(x)\) is equal to zero. To find the x-intercepts from the vertex form, set the function equal to zero and solve for \(x\).

For our function \( f(x)=2(x+2)^{2}-1 \), setting \(0 = 2(x+2)^2 - 1\) and solving for \(x\) gives us the solutions \(-2 \pm \sqrt{1/2}\). These are the x-intercepts, showing where our parabola will touch the x-axis. When plotting these points, we ensure our curve passes through them, creating an accurate depiction of the parabola's intersection with the x-axis.