Problem 34 - 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. \(f(x)=-x^{2}-2 x+3\)

Step-by-Step Solution

Answer

The function \(f(x) = -x^2 - 2x + 3\) can be rewritten as \(f(x) = - (x+1)^2 + 4\), by completing the square. The vertex of the function is at (-1, 4). There is a vertical reflection, horizontal shift of 1 unit to the left, and vertical shift of up 4 units. The x-intercepts can be found by solving \(0 = -(x+1)^2+4\), and the y-intercept is found by evaluating f(0), which is 3. Finally, plot the vertex, intercepts, and apply the shifts and reflection to create the graph of the function \(f(x) = - (x+1)^2 + 4\).

Step 1: Complete the square

To rewrite the function \(f(x) = -x^2 - 2x + 3\) in the form \(f(x) = a(x-h)^2 + k\), we need to complete the square: 1. Group the quadratic and linear terms together: \(-x^2 - 2x\) and leave the constant term: \(+3\). 2. Factor out the coefficient of the quadratic term from the grouped terms: \(-1(x^2 + 2x)\). 3. Add \(\big(\frac{1}{2} \times \text{coefficient of the linear term}\big)^2\) inside the parenthesis and subtract its value outside the parenthesis (we need to add and substract the same value to keep the equation equal): \(-1(x^2 + 2x + 1) + 3 -1(-1)\). 4. Now rewrite the quadratic and linear terms as a perfect square: -1(x+1)^2 + 3 + 1. Now the function is in the form \(f(x) = a(x-h)^2 + k\): \(f(x) = - (x+1)^2 + 4\)

Step 2: Identify the vertex, a-value, and shifts

The vertex form of the function is \(f(x) = - (x+1)^2 + 4\). Comparing it to the standard form, we can now identify the following parts: 1. Vertex (h, k): \((-1, 4)\) 2. A-value, a: -1 (indicates a vertical reflection) 3. Horizontal shift: Left by 1 unit (the opposite sign of h) 4. Vertical shift: Up by 4 units (k-value)

Step 3: Find the intercepts and graph

To find the x-intercepts, set the function equal to 0 and solve for x: \(0 = - (x+1)^2 + 4\) To find the y-intercept, let x be 0 and calculate the function: \( f(0) = - (0+1)^2 + 4 \) Now, we can graph the function based on the vertex, intercepts, and shifts: Vertex: (-1, 4) A-value: -1 (Reflect across x-axis) Horizontal shift: Left by 1 unit Vertical shift: Up by 4 units X-intercepts: Solve for x Y-intercept: f(0) Plot the vertex, intercepts, and apply the shifts and reflection to create the graph of the function \(f(x) = - (x+1)^2 + 4\).

Key Concepts

Quadratic Functions

A quadratic function is a fundamental concept in algebra that represents expressions in the form \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants and \( a eq 0 \). This equation describes a parabola when graphed on the coordinate plane. It is important to recognize that the value of \( a \) determines the direction of the parabola:

• If \( a \) is positive, the parabola opens upwards.
• If \( a \) is negative, the parabola opens downwards.

Understanding the properties of quadratics is essential in solving and graphing these equations. They appear in various applications, from physics to business, where they often model behaviors like the trajectory of objects or economic profit functions.

Vertex Form

The vertex form of a quadratic function is useful when you need to easily identify the parabola's vertex. This form is written as \( f(x) = a(x-h)^2 + k \). Here, \( (h, k) \) represents the vertex of the parabola.

• \( h \) indicates the horizontal shift from the origin.
• \( k \) indicates the vertical shift.
• \( a \) tells you about the parabola's direction and width: negative values flip the parabola, while larger absolute values indicate a narrower shape.

The vertex form is particularly advantageous for graphing because it provides an immediate insight into the location and shape of the parabola. It's also the ideal form for completing the square, which transforms a standard form equation into vertex form for easier manipulation.

Graphing Quadratics

Graphing quadratic functions involves plotting the parabola on a coordinate plane based on its vertex, direction, and width. Here's a step-by-step guide to graphing:

• Start by identifying the vertex \( (h, k) \), which gives the highest or lowest point of the parabola.
• Next, examine the \( a \)-value for directionality: reflect over the x-axis if \( a \) is negative.
• Determine the intercepts for more points; the y-intercept can be found by setting \( x = 0 \), while the x-intercepts require solving the equation \( a(x-h)^2 + k = 0 \).

Once you have these points, sketch the parabola. Importantly, parabolas are symmetric around a vertical line through the vertex, which aids in their plotting. Knowing how to graph quadratic functions is essential for visualizing solutions and understanding the behavior of quadratic models.

Algebra Techniques

Completing the square is a pivotal algebra technique used to solve and manipulate quadratic functions. It's especially useful for converting from standard form to vertex form. Here's how this process works:

• Start by isolating the quadratic and linear terms. If necessary, factor out the leading coefficient.
• Calculate \( \left( \frac{b}{2} \right)^2 \) where \( b \) is the coefficient of the linear term, and add and subtract it inside the equation to complete the square.
• Reformulate the equation into a perfect square, thus converting it into the vertex form \( (x-h)^2 \).

This method simplifies solving quadratics by making it easier to identify the vertex and thus to graph the function. Additionally, it's critical in solving quadratic equations analytically for various mathematics problems, enhancing your algebraic manipulation skills.