Problem 35 - 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. \(y=-x^{2}+6 x-10\)

Step-by-Step Solution

Answer

The function can be rewritten as \(f(x) = -(x-3)^2 - 19\). To graph it, first plot the vertex at (3, -19), which is also the axis of symmetry at \(x=3\). The x-intercepts are \(x = 3 + \sqrt{19}\) and \(x = 3 - \sqrt{19}\), and the y-intercept is (0, -28).

Step 1: Identify a, b, and c in the given function

The given function is \(y = -x^2 + 6x - 10\). Comparing this to the standard quadratic function \(y=ax^2 + bx + c\), we can see that: a = -1, b = 6, and c = -10.

Step 2: Complete the square

To rewrite the function in the form \(f(x) = a(x-h)^2 + k\), we need to complete the square: 1. Divide the coefficient of the linear term (b) by 2, and then square the result: \(\left(\frac{b}{2}\right)^2 = \left(\frac{6}{2}\right)^2 = 9\). 2. Add and subtract the value calculated in step 1 to the quadratic: \(y = -x^2 + 6x + 9 - 9 - 10\). 3. Factor the quadratic and simplify: \(y = -(x^2 - 6x + 9) - 19 = -(x - 3)^2 - 19\). Now we have the function in the desired form: \(f(x) = a(x-h)^2 + k = -(x-3)^2 - 19\), where a = -1, h = 3, and k = -19.

Step 3: Plot the graph and find intercepts

To graph the function, we need to identify the vertex, axis of symmetry, and intercepts: 1. Vertex: The vertex of the function is given by the coordinates (h, k) = (3, -19). 2. Axis of symmetry: The axis of symmetry is a vertical line that passes through the x-coordinate of the vertex: \(x = h = 3\). 3. X-intercepts: To find the x-intercepts, set y = 0 and solve for x: \(0 = -(x-3)^2 - 19\) \(-(x-3)^2 = 19\) \(x-3 = \pm\sqrt{19}\) The x-intercepts are \(x = 3 + \sqrt{19}\) and \(x = 3 - \sqrt{19}\). 4. Y-intercept: To find the y-intercept, set x = 0 and solve for y: \(y = -(0-3)^2 - 19 = -9 - 19 = -28\) The y-intercept is (0, -28). Now, with all the points and necessary information, the graph of the function can be drawn. The graph will have the vertex at (3, -19), the axis of symmetry at \(x=3\), and will pass through the x-intercepts at \(x = 3 + \sqrt{19}\) and \(x = 3 - \sqrt{19}\) and the y-intercept at (0, -28).

Key Concepts

Quadratic Functions

Quadratic functions are equations that take the form \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, and \( x \) is the variable. These functions describe a wide range of real-world scenarios, from projectile motion to economics.
In the given function, \( y = -x^2 + 6x - 10 \), it's classified as a quadratic function because it includes an \( x^2 \) term. The coefficient \( a = -1 \) tells us that the parabola will open downward, as negative coefficients cause this inversion.

Quadratic functions are crucial in mathematics because they model non-linear relationships and provide a visual representation in the form of parabolas. Each function has a unique shape and behavior, based on its coefficients.

Parabolas

Parabolas are the graphs of quadratic functions. They have a characteristic 'U' shape that can open upwards or downwards, depending on the sign of the coefficient \( a \). In our function \( y = -x^2 + 6x - 10 \), the parabola opens downward because the coefficient \( a = -1 \) is negative.
A parabola has several key features:

• Vertex: The highest or lowest point on the parabola, determining its maximum or minimum value. In our example, the vertex is at \((3, -19)\).
• Axis of symmetry: A vertical line that divides the parabola into two mirror-image halves; for our function, it's \(x = 3\).
• Intercepts: Points where the parabola crosses the axes. The y-intercept is \((0, -28)\), and the x-intercepts are \(x = 3 \pm \sqrt{19}\).

Understanding these elements helps visualize the function and predict its behavior under different conditions.
Parabolas find use in many fields, such as in designing satellite dishes or reflecting telescopes, where this shape is essential due to its reflective properties.

Vertex Form

The vertex form of a quadratic function is expressed as \( f(x) = a(x-h)^2 + k \), where \((h, k)\) are the coordinates of the vertex. This form is particularly useful as it immediately reveals the vertex of the parabola and thus helps in graphing the function easily.
By completing the square for \( y = -x^2 + 6x - 10 \), we transform it into its vertex form: \( f(x) = -(x-3)^2 - 19 \). Here, \( h = 3 \) and \( k = -19 \), making the vertex \((3, -19)\).

This transformation is done by manipulating the quadratic equation to make \( (x-h)^2 \) a part of it, which converts a standard quadratic equation into an easier-to-use form. The vertex form is advantageous for

• Identifying the vertex readily, without additional calculations.
• Seeing how shifts in \( h \) and \( k \) will move the parabola horizontally and vertically.

With this form, analyzing and graphing parabolas become more intuitive, assisting students and professionals alike in visualizing mathematical concepts quickly.