Problem 42 - Beginning and Intermediate Algebra

Question

Rewrite function in the form \(f(x)=a(x-h)^{2}+k\) by completing the square. Then, graph the function. Include the intercepts. \(f(x)=x^{2}+5 x+\frac{21}{4}\)

Step-by-Step Solution

Answer

The quadratic function in vertex form is \(f(x) = (x + \frac{5}{2})^2 - 1\). The vertex is at \((-2.5, -1)\), with axis of symmetry \(x = -2.5\). The x-intercepts are approximately \((-1.07, 0)\) and \((-4.93, 0)\), and the y-intercept is at \((0,\frac{21}{4})\). To graph the function, plot these points and draw a parabola through them with the vertex as the minimum point.

Step 1: Identify the quadratic function

The given quadratic function is \(f(x)=x^{2}+5x+\frac{21}{4}\).

Step 2: Complete the square

In order to rewrite \(f(x)\) in the vertex form, we need to complete the square. To do this, we'll take the quadratic function, \(f(x)=x^{2}+5x+\frac{21}{4}\), and rewrite it as follows: \(f(x)=x^{2}+5x+\frac{25}{4}-\frac{25}{4}+\frac{21}{4}\) The middle term (5x) is divided by 2 and then squared, to give us \(\left(\frac{5}{2}\right)^2 = \frac{25}{4}\). We then add and subtract \(\frac{25}{4}\) in the function.

Step 3: Rewrite the function in vertex form

Now, we can rewrite the function as: \(f(x)=(x^2+5x+\frac{25}{4})-\frac{25}{4}+\frac{21}{4}\) As a next step, we will factor the perfect square trinomial inside the parentheses: \(f(x)=(x+\frac{5}{2})^2-\frac{4}{4}\) So, we have found the vertex form of the given quadratic function: \(f(x)=(x+\frac{5}{2})^2-\frac{4}{4}\).

Step 4: Identify the vertex and the axis of symmetry

The vertex of the quadratic function is given by the coordinates (h, k), where h and k are obtained from the function \(f(x)=a(x-h)^2+k\). In our case, the vertex is \(\left(-\frac{5}{2}, -1\right)\), and the axis of symmetry is the vertical line \(x=-\frac{5}{2}\).

Step 5: Find x and y intercepts

To find the x-intercepts, set \(f(x)=0\) and solve for x: \(0 = (x+\frac{5}{2})^2-\frac{4}{4}\) Solving this equation gives us x-intercepts at (x≈-1.07) and (x≈-4.93). To find the y-intercept, set \(x=0\) and solve for f(x): \(f(0) = (0+\frac{5}{2})^2-\frac{4}{4}\) \(f(0) = \frac{21}{4}\) So, the y-intercept is at the point \((0,\frac{21}{4})\).

Step 6: Graph the function

To graph the function, we will first plot the vertex \(\left(-\frac{5}{2}, -1\right)\), the axis of symmetry \(x=-\frac{5}{2}\), and the intercepts \((0,\frac{21}{4})\), \((x≈-1.07)\), and \((x≈-4.93)\). Then, we draw a parabola through these points with the vertex representing the minimum point.

Key Concepts

Quadratic Function

A quadratic function is a type of polynomial function specifically characterized by having an equation of the form \( ax^2 + bx + c \), where \( a eq 0 \). To fully understand quadratic functions, here are some key points:

• The general shape of the graph of a quadratic function is a parabola.
• Parabolas can open upwards or downwards based on the sign of the leading coefficient \( a \). If \( a > 0 \), the parabola opens upwards. If \( a < 0 \), it opens downwards.
• Quadratic functions display symmetry; this is a crucial aspect that simplifies graphing and understanding them.

The function from our exercise is \( f(x)=x^2+5x+\frac{21}{4} \), a typical quadratic expression falling under this category. This initial form helps us recognize it as a quadratic function and start analyzing its features, such as the vertex and intercepts.

Vertex Form

The vertex form of a quadratic function makes it easier to identify the vertex of the parabola. This form is written as \( f(x) = a(x-h)^2+k \), wherein \((h, k)\) represents the vertex.

• The vertex is the point where the parabola changes direction, either reaching a maximum or a minimum.
• Converting a quadratic function to vertex form involves completing the square, a method that helps reveal the vertex and simplify graphing.

In our example, we transformed \( f(x)=x^2+5x+\frac{21}{4} \) into its vertex form \( f(x)=(x+\frac{5}{2})^2-1 \). Now, you can easily see that the vertex of this parabola is \(-\frac{5}{2}, -1\), and use this information for graphing and analysis.

X and Y Intercepts

The x and y intercepts of a quadratic function are crucial for sketching its graph accurately.

• X-intercepts are points where the graph crosses the x-axis. They occur when \( f(x) = 0 \).
• To find them, solve \((x+h)^2 + k = 0\). For our function, the solutions are the x-intercepts: approximately \( x \approx -1.07 \) and \( x \approx -4.93 \).
• Y-intercepts appear where the graph crosses the y-axis, determined by evaluating \( f(0) \).
• In our exercise, the y-intercept is \( \left(0, \frac{21}{4}\right) \), showing where the parabola touches the y-axis.

These intercepts help outline the boundaries and essential points of our graph, aiding in the creation of an accurate visual representation of the function.

Axis of Symmetry

The axis of symmetry of a quadratic function provides insight into its mirrored balance, playing a critical role in understanding its behavior.

• The axis of symmetry is a vertical line passing through the vertex of the parabola, specifically at \( x = h \). It divides the parabola into two symmetrical halves.
• In our example, with the vertex located at \( \left(-\frac{5}{2}, -1\right) \), the axis of symmetry is \( x = -\frac{5}{2} \).
• This symmetry ensures that for every point on one side of the axis, there is a corresponding point on the opposite side.

Knowing the axis helps in predicting the behavior of the function and constructing accurate graphs. It allows visual confirmation that the parabola is drawn correctly by checking the symmetry relative to this vertical line.