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Magazine Chapter 3: Problem 12
Write each quadratic function in the form \(y=a(x-h)^{2}+k\) and sketch its graph. $$y=x^{2}+5 x$$
Short Answer
Expert verified
The vertex form of the function is $$y = (x + \frac{5}{2})^2 - \frac{25}{4}$$ The vertex is at $$(-\frac{5}{2}, -\frac{25}{4})$$ and the parabola opens upwards.
Step by step solution
01
Identify the quadratic function
The given quadratic function is $$y = x^2 + 5x$$
02
Complete the square
To convert the quadratic function into the vertex form, complete the square. Start by rewriting the equation: $$y = x^2 + 5x$$ Add and subtract $$\frac{(5)}{2}^2 = \frac{25}{4}$$ inside the quadratic expression: $$y = x^2 + 5x + \frac{25}{4} - \frac{25}{4}$$
03
Rewrite as a square of a binomial
Now, express the perfect square trinomial as a binomial square: $$y = (x + \frac{5}{2})^2 - \frac{25}{4}$$ Thus, the function in vertex form is $$y = 1 (x - (- \frac{5}{2}))^2 - \frac{25}{4}$$
04
Identify the vertex and sketch the graph
In the form $$y = a(x - h)^2 + k$$ we found $$a=1, h = -\frac{5}{2}, k = -\frac{25}{4}$$ The vertex of the parabola is at $$(-\frac{5}{2}, -\frac{25}{4})$$ Since the coefficient of $$x^2$$ is positive, the parabola opens upwards. The vertex is its minimum point.
05
Plot key points and sketch the graph
The graph can be sketched by plotting key points around the vertex $$(-\frac{5}{2}, -\frac{25}{4})$$ Use a few values of $$x$$ near $$x=-\frac{5}{2}$$ to get corresponding values of $$y$$ and plot the points. Draw a smooth curve through these points.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Completing the Square
Completing the square is a method used to transform a quadratic equation into its vertex form. This process helps us identify key features of the quadratic function easily. In this exercise, we start with the quadratic function given as \(y = x^2 + 5x\). The goal is to rewrite this in the form \(y = a(x-h)^2 + k\), which is the vertex form.
Step-by-Step Process:
Step-by-Step Process:
- First, we need to make the quadratic expression a perfect square trinomial. This can be done by adding and subtracting a specific value inside the equation.
- The value to add and subtract is \(\left(\frac{b}{2}\right)^2\), where \(b = 5\).
- Thus, we add and subtract \(\left(\frac{5}{2}\right)^2 = \frac{25}{4}\) inside the quadratic term: \(y = x^2 + 5x + \frac{25}{4} - \frac{25}{4}\).
- This changes the expression into a perfect square trinomial \(x^2 + 5x + \frac{25}{4}\), which can be rewritten as \(\left(x + \frac{5}{2}\right)^2\).
- Don't forget to include the subtracted value \(\frac{25}{4}\) after the binomial: \(y = \left(x + \frac{5}{2}\right)^2 - \frac{25}{4}\).
- Now, we have the equation in the desired vertex form \(y = \left(x - (-\frac{5}{2})\right)^2 - \frac{25}{4}\).
Understanding Quadratic Functions
Quadratic functions are polynomial functions of degree 2, typically represented as \(y = ax^2 + bx + c\). These functions have a characteristic U-shaped graph called a parabola. Understanding how to convert these into vertex form \(y = a(x-h)^2 + k\) makes it easier to analyze and graph them.
Key Characteristics:
Vertex Form:
Key Characteristics:
- The general form of a quadratic function is \(y = ax^2 + bx + c\).
- The graph of a quadratic function is a parabola.
- If \(a\) is positive, the parabola opens upwards; if \(a\) is negative, it opens downwards.
Vertex Form:
- The vertex form of the quadratic function is \(y = a(x-h)^2 + k\).
- The vertex \((h,k)\) gives the turning point of the parabola.
- Converting to vertex form is particularly useful for graphing because it gives you the vertex directly, which is a key feature of the graph.
Finding the Parabola Vertex
The vertex of the parabola is the point where it turns. In a quadratic function in vertex form \(y = a(x-h)^2 + k\), the vertex is given directly by the point \((h, k)\).
Steps to Identify the Vertex:
Graphing the Quadratic Function:
Steps to Identify the Vertex:
- Once you've completed the square and rewritten the quadratic equation in vertex form, you can easily identify the vertex.
- For the quadratic function \(y = x^2 + 5x\), after completing the square, we get \(y = \left(x + \frac{5}{2}\right)^2 - \frac{25}{4}\).
- From this, we can see that \(h = -\frac{5}{2}\) and \(k = -\frac{25}{4}\).
Graphing the Quadratic Function:
- Plot the vertex point first. In this case, the vertex is \((-\frac{5}{2}, -\frac{25}{4})\).
- Check the sign of the coefficient of \(x^2\) to determine the direction of the parabola. Here, \(a = 1\) (positive) means the parabola opens upwards.
- Plot additional points on either side of the vertex by substituting \(x\) values to find corresponding \(y\) values.
- Draw a smooth curve through these points to complete the graph.
