Problem 89 - Algebra and Trigonometry Real Mathematics, Real People

Question

Sketch the graph of the equation by hand. Verify using a graphing utility. $$y=(x-3)^{2}-7$$

Step-by-Step Solution

Answer

The vertex of the graph is (3, -7), the graph opens upwards, and the y-intercept is at (0,2). It's a parabola opening upwards with vertex as its lowest point as verified by a graphing utility.

Step 1: Identify the Vertex

The vertex form of a quadratic function is \(y=a(x-h)^{2}+k\). Here, the vertex is given by (h, k), and in the equation \(y=(x-3)^{2}-7\), that would equate to (3, -7). So, point at (3, -7) on the graph would be the vertex of the function.

Step 2: Determine the Opening Direction

In a quadratic function, if the coefficient 'a' is greater than 0, the graph opens upwards and if 'a' is less than 0, the graph opens downwards. In the function \(y=(x-3)^{2}-7\), the coefficient 'a' is 1, which is greater than 0. Therefore, the graph of the function opens upwards.

Step 3: Find the y-intercept

The y-intercept is the point where the graph intersects the y-axis. For this, we substitute x=0 in the function. So, \(y=(0-3)^{2}-7 = 9-7 = 2\). The graph intersects the y-axis at the point (0,2).

Step 4: Sketch the Graph

Use the information from the previous steps, i.e., the vertex, opening direction, and y-intercept, to sketch the graph. Begin by plotting the vertex at point (3,-7) and the y-intercept at point (0,2). Since our parabola opens upwards, sketch a symmetric curve upward from the vertex, which should pass through the y-intercept.

Step 5: Verify using a graphing utility

Finally, verify the hand-drawn graph using a graphing utility. It is expected that the utility's graph should match with the hand-drawn sketch, having its vertex at (3, -7) and y-intercept at (0,2) and opening upwards.

Key Concepts

Vertex Form

The vertex form of a quadratic function helps us easily identify the vertex of a parabola, which is a crucial part in sketching its graph. A quadratic equation in vertex form is expressed as:

• \( y = a(x-h)^2 + k \)

Here, the vertex is the point \((h, k)\). For the equation \( y = (x-3)^2 - 7 \), it becomes evident that the vertex is \((3, -7)\). The vertex gives us the exact turning point of the parabola. This makes it easier to sketch the graph accurately. Understand that the "\(h\)" value moves the vertex left or right while "\(k\)" moves it up or down. This representation is particularly useful for understanding and predicting the behavior of the graph.

Parabola

A parabola is the U-shaped curve represented by a quadratic equation. Its direction depends on the coefficient \(a\) in the vertex form.
- If \(a\) is positive, the parabola opens upwards, which looks like a smiling face.- If \(a\) is negative, it opens downwards, resembling a frown.
In our equation \( y = (x-3)^2 - 7 \), the coefficient \(a\) is 1, which means the parabola opens upwards.
After determining its direction, you can use points like the vertex and the y-intercept to sketch its path efficiently. Remember that the parabola is symmetric around a vertical line through the vertex, aiding in constructing an accurate graph.

Graphing Utility

Validating your hand-drawn graph with a graphing utility provides assurance of accuracy. A graphing utility is a tool or software that visualizes mathematical equations instantly.
To verify, simply input the equation \( y = (x-3)^2 - 7 \). The graph should reflect:

• A vertex at (3, -7)
• Opening upwards
• Y-intercept at (0, 2)

The utility serves as a double-check mechanism to ensure all calculations are correct. It’s a great way to see the effects of different transformations and comprehend their impact on the curve's shape and position.

y-intercept

The y-intercept is where the parabola crosses the y-axis. This is critical for plotting additional points on the graph. To find it, substitute \(x = 0\) in the equation.
For \(y = (x-3)^2 - 7\), calculating yields:

• Substitute: \((0-3)^2 - 7 = 9 - 7 = 2\)
• Y-intercept: (0, 2)

The y-intercept provides a point on the graph, helping to establish the curve's position. Combined with the vertex and parabola's symmetry, it allows us to accurately sketch the graph.