Problem 33 - Elementary Algebra

Question

Construct the graphs of the following equations. $$ y=(x-2)^{2}+3 $$

Step-by-Step Solution

Answer

Answer: The vertex of the parabola is (2, 3), and it opens upwards.

Step 1: Identify the vertex

The vertex of a parabola given by the equation y = (x - h)^2 + k is (h, k). In our equation, y = (x - 2)^2 + 3, we have h = 2 and k = 3. Therefore, our vertex is at the point (2, 3).

Step 2: Determine the parabola's orientation

Since our equation y = (x - 2)^2 + 3 has a squared term with a positive coefficient, the parabola will open upwards. This means its minimum value will be at the vertex, and the graph will extend upwards on both sides of the vertex.

Step 3: Plot the vertex and find additional points

Start by plotting the vertex on the coordinate plane at point (2, 3). To find additional points for our parabola, we can choose a few x-values and compute the corresponding y-values using our equation (x-2)^2+3. For example, let's choose x = 0, 1, 3, and 4. When x = 0, y = (0 - 2)^2 + 3 = 7 When x = 1, y = (1 - 2)^2 + 3 = 4 When x = 3, y = (3 - 2)^2 + 3 = 4 When x = 4, y = (4 - 2)^2 + 3 = 7 Now we have the points (0, 7), (1, 4), (3, 4), and (4, 7) to plot on our graph in addition to the vertex.

Step 4: Connect the points and create the graph

Finally, plot the points we found in Step 3 on the coordinate plane along with the vertex. These points should help us see the symmetrical shape of the parabola. Connect the points with a smooth curve, making sure that the curve opens upwards and passes through each point, including the vertex. This will give you the graph of the equation y = (x - 2)^2 + 3.

Key Concepts

Vertex Form

In mathematics, understanding the vertex form of a quadratic equation is crucial when graphing parabolas. The general formula for a parabola in vertex form is \( y = (x - h)^2 + k \). Here, \( h \) and \( k \) are constants that identify the vertex's coordinates. The vertex coordinates are extremely valuable because they tell us the exact point where the parabola either reaches a minimum or maximum value.

• \( h \) determines the horizontal shift.
• \( k \) impacts the vertical shift.

Considering our equation \( y = (x - 2)^2 + 3 \), the vertex is at (2, 3). This means the parabola shifts 2 units to the right and 3 units up from the origin on the graph. The vertex thus shows not only the turning point but also helps set the stage for the parabola's overall shape on the graph.

Quadratic Functions

Quadratic functions are polynomial functions of degree 2, represented by the standard form \( y = ax^2 + bx + c \), or by the vertex form regarded here. They create smooth "U" shaped graphs known as parabolas. The quadratic function's nature allows it to model various real-world phenomena, such as projectile motion or optimizing space.When we say a quadratic function or equation, the most distinctive characteristic is its squared term \( x^2 \). This term shapes how the curve opens. In vertex form, when the coefficient of this term, often denoted as \( a \), is positive, the parabola opens upwards. If \( a \) were negative, it would open downwards, creating a maximum point at the vertex. These shapes and forms help math students visualize and understand the behavior of quadratic equations.

Coordinate Plane

To graph a parabola, we employ the coordinate plane. This fundamental graphing tool comprises two perpendicular lines: the x-axis (horizontal) and the y-axis (vertical). The point at which they intersect is the origin, denoted by (0, 0).Each point on this plane is expressed as an ordered pair \( (x, y) \). The x-axis informs us how far left or right we are, while the y-axis indicates the vertical distance.

• The coordinate plane is divided into four quadrants.
• Understanding the quadrants assists in navigating the plane while plotting points.

In our exercise, using the coordinate plane involved first placing our vertex at (2, 3). This point becomes the central guide for accurately plotting the entire parabola.

Plotting Points

Plotting points is a fundamental step when drawing any graph, including our parabola. It involves placing each coordinate pair, like \( (x, y) \), onto the coordinate plane accurately. In doing so, we create a visual representation of the mathematical equation.For the parabola \( y = (x - 2)^2 + 3 \), we began by identifying and plotting the vertex (2, 3). From there, additional points were calculated by choosing various x-values and solving for their corresponding y-values.

• \( x = 0, y = 7 \) forms the point (0, 7).
• \( x = 1, y = 4 \) forms the point (1, 4).
• \( x = 3, \, y = 4 \) forms the point (3, 4).
• \( x = 4, y = 7 \) forms the point (4, 7).

Plot these points and then join them in a smooth curve to reveal the shape of the parabola. This visualization emphasizes the symmetry of the parabola and its progression from the vertex.