Problem 18 - Algebra 2

Question

Graph each function. $$ y=x^{2}-8 x+18 $$

Step-by-Step Solution

Answer

The graph is a parabola with vertex at (4, 2) and opens upwards.

Step 1: Identify the Vertex Form

The given function is a quadratic equation in the standard form \(y = ax^2 + bx + c\), with \(a = 1\), \(b = -8\), and \(c = 18\). To graph the function, first, convert it into vertex form \(y = a(x - h)^2 + k\), where \((h, k)\) is the vertex of the parabola.

Step 2: Complete the Square

To convert \(y = x^2 - 8x + 18\) into vertex form, complete the square:1. Take the coefficient of \(x\), which is -8, divide it by 2, giving -4, and square it to get 16.2. Add and subtract this square inside the equation: \(y = (x^2 - 8x + 16) + 18 - 16\).3. Simplify to get: \(y = (x - 4)^2 + 2\).Thus, the vertex form is \(y = (x - 4)^2 + 2\).

Step 3: Identify Key Features

From the vertex form \(y = (x - 4)^2 + 2\), identify that the vertex of the parabola is \((4, 2)\). The parabola opens upwards since the coefficient of \((x-4)^2\) is positive.

Step 4: Plot the Vertex

Plot the vertex \((4, 2)\) on the Cartesian plane. This point is the peak of the parabola and determines the symmetry axis, which is the vertical line \(x = 4\).

Step 5: Determine Additional Points

Choose other x-values to find more points on the graph. For instance:- If \(x=3\), then \(y = 9\).- If \(x=5\), then \(y = 9\).These points are symmetric about the line \(x = 4\).

Step 6: Sketch the Parabola

Using the vertex and additional points, sketch the parabola. Draw a smooth curve connecting all the plotted points, ensuring that it has a U-shape opening upwards.

Key Concepts

Completing the Square

Completing the square is a technique used to rewrite quadratic equations in a way that makes them easier to solve or graph. This method is particularly useful for converting a quadratic equation from its standard form into vertex form.
First, identify the coefficient of the term with the linear variable. In the quadratic equation \(y = x^2 - 8x + 18\), the linear coefficient is \(-8\). Follow these steps to complete the square:

• Divide the linear coefficient by 2. This gives you \(-4\).
• Square the result. \((-4)^2 = 16\).
• Add and subtract this square inside the equation to maintain equality: \(y = (x^2 - 8x + 16) + 18 - 16\).

This process transforms the quadratic into a perfect square trinomial: \((x - 4)^2\).
Completing the square highlights the vertex form, which is essential for finding the vertex of the parabola quickly.

Standard Form of Quadratic Equation

The standard form of a quadratic equation is given by \(y = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants. In this form, \(x\) represents the variable.
The equation \(y = x^2 - 8x + 18\) is a classic example of a quadratic in standard form with \(a = 1\), \(b = -8\), and \(c = 18\).
This form is useful because it provides immediate information about the parabola:

• It tells us the direction of the parabola's opening. If \(a > 0\), the parabola opens upwards, and if \(a < 0\), it opens downwards.
• The constants \(b\) and \(c\) help determine other features such as the axis of symmetry and the y-intercept.

While the standard form is excellent for calculation, conversion to vertex form can make understanding the graph's shape more intuitive.

Graphing Quadratic Functions

Graphing a quadratic function involves plotting points to reveal the shape of the parabola. The vertex form of the equation, \(y = a(x - h)^2 + k\), makes finding the vertex straightforward.
For \(y = (x - 4)^2 + 2\), the vertex is \((4, 2)\), which is the highest or lowest point on the graph and determines the axis of symmetry, a vertical line through \(x = 4\).
Follow these steps for graphing:

• Start by plotting the vertex \((4, 2)\). This point helps establish the graph's direction and central point.
• Determine the direction of the parabola. Since \(a = 1\) in this example, the parabola opens upwards.
• Select additional \(x\)-values to calculate corresponding \(y\)-values, ensuring they are symmetrical about the vertex. For example, with \(x = 3\) and \(x = 5\), both produce \(y = 9\).
• Plot these points and draw a smooth curve through them, ensuring a U-shape that opens upward.

Graphing enables visualization of the quadratic function, showing clearly how it changes and revealing key characteristics like symmetry and vertex location.