Problem 38 - 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. \(y=2 x^{2}-8 x+2\)

Step-by-Step Solution

Answer

The function after completing the square is \(f(x) = 2(x - 2)^2 - 2\). The x-intercepts are \((1, 0)\) and \((3, 0)\), and the y-intercept is \((0, 6)\). To graph the function, plot the vertex \((2, -2)\), axis of symmetry \(x = 2\), and the intercepts, and draw the parabola opening upwards.

Step 1: Complete the square

Rewrite the given function \(y = 2x^2 - 8x + 2\) in the form \(f(x) = a(x - h)^2 + k\): 1. Factor out the coefficient of the \(x^2\) term from the first two terms: \[y = 2(x^2 - 4x) + 2\] 2. Add and subtract the square of half of the coefficient of the x-term inside the parenthesis: \[y = 2(x^2 - 4x + 4 - 4) + 2\] 3. Simplify and rewrite the function in the desired form: \[y = 2[(x - 2)^2 - 2] + 2\] \[y = 2(x - 2)^2 - 2\] So, the function after completing the square is \(f(x) = 2(x - 2)^2 - 2\).

Step 2: Find the intercepts

1. x-intercepts (when y=0): Set \(y = 0\) in the function: \[0 = 2(x - 2)^2 - 2\] Add 2 to both sides: \[2 = 2(x - 2)^2\] Divide both sides by 2: \[1 = (x - 2)^2\] Take the square root: \[x - 2 = \pm 1\] Solve for x: \[x = 2 \pm 1\] So, we have two x-intercepts: \((1,0)\) and \((3,0)\). 2. y-intercept (when x=0): Plug in \(x=0\) in the function: \[y = 2(0 - 2)^2 - 2\] \[y = 2(4) - 2\] \[y = 8 - 2\] \[y = 6\] The y-intercept is \((0, 6)\).

Step 3: Graph the function

To graph \(f(x) = 2(x - 2)^2 - 2\), we need the following information: - Vertex: \((h, k) = (2, -2)\) - Axis of symmetry: \(x = h = 2\) - x-intercepts: \((1, 0)\) and \((3, 0)\) - y-intercept: \((0, 6)\) Now, you can plot these points and draw the parabola with the given characteristics. The vertex is the bottom-most point, and the parabola opens upwards due to the positive coefficient of the \((x-2)^2\) term. The axis of symmetry is the vertical line x=2 that divides the parabola equally. The x-intercepts are the points where the parabola crosses the x-axis, and the y-intercept is where it crosses the y-axis.

Key Concepts

Quadratic Functions

Quadratic functions are mathematical expressions of the form \( ax^2 + bx + c \). These functions, defined by a second-degree polynomial, are essential in algebra and calculus. They describe a parabola when graphed on a Cartesian plane.

• The general form is \( ax^2 + bx + c \), where \( a eq 0 \).
• If \( a > 0 \), the parabola opens upwards, like a smile.
• If \( a < 0 \), it opens downwards, resembling a frown.

The constant \( a \) controls the width and direction, \( b \) affects the position, and \( c \) is the y-intercept. Quadratic functions are at the heart of many real-world phenomena, such as projectile motion.

Graphing Parabolas

Graphing parabolas involves plotting the graph of a quadratic function. A quadratic function's graph is a symmetrical curve called a parabola. Understanding the shape and position of the parabola helps in interpreting the function.

• The vertex is the turning point of the parabola.
• It has an axis of symmetry, a vertical line that passes through the vertex.
• Intercepts are where the parabola crosses the axes.

Graphing involves the following steps: - Identify the vertex. - Determine the axis of symmetry. - Plot intercepts. - Use additional points to sketch the entire parabola. By following these steps, we can visualize the quadratic function effectively.

Vertex Form of a Quadratic Function

The vertex form expresses a quadratic function in a way that makes identifying the vertex straightforward. This form is particularly useful for graphing and understanding the parabola's properties.The vertex form is given by:\[ f(x) = a(x - h)^2 + k \]Where:- \((h, k)\) is the vertex.- \(a\) indicates the direction and width.To convert a quadratic equation from standard form to vertex form, you can use the technique of completing the square. This involves:- Factoring out \(a\) from the first two terms if it's not 1.- Completing the square inside the parentheses.- Adjusting constant terms outside the parentheses.Using this form makes it easy to graph quadratics and see transformations like shifts and stretches.

X-intercepts and Y-intercepts

Intercepts are crucial points where the graph of a function crosses the axes. They help in understanding the behavior of the function at these intersections.

• X-intercepts: These are points where the graph crosses the x-axis. They occur when \( y = 0 \). Solving the equation for \( x \) gives the x-intercepts.
• Y-intercepts: This is the point where the graph crosses the y-axis. It occurs when \( x = 0 \). Substituting \( x = 0 \) into the equation provides the y-intercept.

For the equation \(y = 2(x - 2)^2 - 2\):- X-intercepts are found by setting \(y = 0\).- The Y-intercept is found by setting \(x = 0\).Recognizing these intercepts allows us to plot important reference points on the graph of the parabola.