Problem 33 - 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. $$ f(x)=x^{2}+6 x+3 $$

Step-by-Step Solution

Answer

The axis of symmetry is \(x = -3\). The domain is all real numbers. The range is \(y \geq -6\).

Step 1: Identify the vertex of the graph

By completing the square, write the function \(f(x) = x^2 + 6x + 3\) in vertex form, which is \(f(x) = (x - h)^2 + k\), where \( (h,k)\) is the vertex of the parabola. Now, set \(a = 1\), \(b = 6\), and \(c = 3\). Thus, \(h = -\frac{b}{2a} = -\frac{6}{2(1)} = -3\) and \(k = f(h) = (-3)^2 + 6(-3) + 3 = -6\). Therefore, the vertex is \((-3, -6)\).

Step 2: Find the x-intercepts (if any) of the graph

To find the x-intercepts, set \(f(x) = 0\). So \(x^{2}+6 x+3=0\). Factoring this yields \((x+3)^2 = 0\). Setting the factored form equal to zero, we find the solutions to be \(-3\) and \(-3\) (i.e., \(-3\) is a double root). Therefore, the x-intercepts are \((-3, 0)\).

Step 3: Draw the sketch of the graph

Because the coefficient of \(x^{2}\) is positive, the graph opens up. By using the vertex found in step one, and the x-intercepts found in step two, plot these points and draw a symmetrical parabola that passes through them.

Step 4: Determine the axis of symmetry, domain, and range of the function

The axis of symmetry of the parabola is at \(x = h = -3\). The domain of the function \(f(x)\) is all real numbers, because \(x\) can take any real number in a quadratic function. The range of the function \(f(x)\) is all real numbers greater than or equal to -6, as the graph opens upwards and the minimum point is at \(y = k = -6\).

Key Concepts

Vertex of a Parabola

Understanding the vertex of a parabola is essential in graphing quadratic functions. The vertex is the highest or lowest point on the parabola, depending on whether it opens upwards or downwards. It represents a pivot point from which the parabola is symmetric. In mathematical terms, the vertex's coordinates are given by \( (h, k) \), where \( h \) is the x-value of the vertex and \( k \) is the corresponding y-value.

In the given function \( f(x) = x^2 + 6x + 3 \), we find the vertex by completing the square and converting the quadratic into the vertex form \( f(x) = (x - h)^2 + k \). Here, the vertex is identified as \( (-3, -6) \), which is crucial for sketching the graph correctly. The vertex is not just a point, but it is also key to determining the axis of symmetry and the range of the function.

X-Intercepts of a Quadratic

The x-intercepts of a quadratic function, also known as the roots or zeros, are the points where the graph crosses the x-axis. To find these intercepts, we set the quadratic equation equal to zero and solve for \( x \).

For \( f(x) = x^2 + 6x + 3 \) we solve \( x^2 + 6x + 3 = 0 \). The solutions, in this case, are both -3, meaning we have a double root. Therefore, the x-intercept of the graph is a single point at \( (-3, 0) \), indicating that the parabola touches the x-axis only at this point. When graphing, this helps us anchor the parabola on the x-axis and further confirms the axis of symmetry.

Axis of Symmetry

The axis of symmetry of a parabola is a vertical line that passes through the vertex and divides the parabola into two mirror images. It's an important feature because it reveals the balanced nature of the parabola and aids in graphing.

For our function \( f(x) = x^2 + 6x + 3 \), the axis of symmetry is the line \( x = -3 \), which means every point on the parabola is reflected across this line. This line is not only a powerful tool for graphing but also significant for analyzing the properties of the quadratic function.

Domain and Range of a Function

The domain of a function refers to the complete set of possible values of the independent variable (usually \( x \)), while the range is the complete set of possible output values (usually \( y \)).

For quadratic functions like \( f(x) = x^2 + 6x + 3 \) the domain is always all real numbers, \( (-\infty, \infty) \), because there is no restriction on the \( x \) values. As for the range, since our given parabola opens upwards and has a vertex at \( (-3, -6) \), the range is all real numbers \( y \) such that \( y \geq -6 \). In other words, the graph extends indefinitely upwards from the minimum point at the vertex.