Problem 44 - Algebra and Trigonometry Real Mathematics, Real People

Question

Write the standard form of the quadratic function that has the indicated vertex and whose graph passes through the given point. Use a graphing utility to verify your result. Vertex: \(\left(-\frac{1}{4},-1\right) ; \quad\) Point: \(\left(0,-\frac{17}{16}\right)\)

Step-by-Step Solution

Answer

The standard form of the quadratic function with a given vertex and passing through the given point is \(f(x) = -1(x + \frac{1}{4})^2 - 1\). This result should be verified using a graphing tool.

Step 1: Write down the general form of the quadratic function

The general form of the quadratic function that shares the vertex \(\left(-\frac{1}{4},-1\right)\) is given by \(f(x) = a(x - (-\frac{1}{4}))^2 - 1\).

Step 2: Substitute the coordinates of the given point into the function

Substitute the x,y values of the point \(\left(0,-\frac{17}{16}\right)\) into the function: \(-\frac{17}{16} = a(0 - (-\frac{1}{4}))^2 - 1\). Simplifying the equation gives \(-\frac{17}{16} + 1 = a(\frac{1}{4})^2\), which further simplifies to \(-\frac{1}{16} = a(\frac{1}{16})\).

Step 3: Solve for \(a\)

To identify the constant \(a\) in the equation, solve the equation from step 2. This results in \(a = -1\).

Step 4: Express the function in standard form

Substitute the resolved \(a\) into the general form of the function. The standard form of the quadratic function becomes: \(f(x) = -1(x + \frac{1}{4})^2 - 1 \).

Step 5: Verify the result with a graphing utility

The written standard form of the function should be verified graphically using a graphing calculator or similar tool. The graph should display a parabola with vertex at \(\left(-\frac{1}{4},-1\right)\), passing through point \(\left(0,-\frac{17}{16}\right)\).

Key Concepts

Vertex Form

When dealing with quadratic functions, one important way to express them is in vertex form. The vertex form allows us to clearly see the vertex of the parabola. The vertex form of a quadratic function is written as:

• \[f(x) = a(x-h)^2 + k\]

Here,

• \((h, k)\) is the vertex of the parabola.
• \(a\) determines the width and direction of the parabola. If \(a > 0\), the parabola opens upwards. If \(a < 0\), it opens downwards.

Knowing the vertex makes it easy to graph the quadratic, as you can plot this point first and then use additional points to sketch the rest of the parabola. In our exercise, the vertex given is \((-\frac{1}{4}, -1)\), which we can plug into the vertex form to start forming the quadratic function. This shows the significance of the vertex form in understanding and graphing quadratic functions.

Standard Form

The standard form of a quadratic function is another way of writing the equation of a parabola. In standard form, a quadratic function looks like:

• \[f(x) = ax^2 + bx + c\]

Each term in this equation has a specific role:

• \(a\) again influences the width and direction of the parabola, similar to its role in the vertex form.
• \(b\) affects the position of the parabola along the x-axis.
• \(c\) determines the y-intercept – where the parabola crosses the y-axis.

In the exercise, after finding the value of \(a\), we converted the equation from the vertex form to the standard form. Standard form is particularly useful for analyzing the roots of the quadratic function by using the quadratic formula and other techniques. Having both forms – vertex and standard – gives you a comprehensive understanding of the quadratic function's behavior.

Graphing Utility

To ensure the accuracy of our quadratic function, a graphing utility can be incredibly helpful. Graphing utilities, such as graphing calculators or computer software, allow us to visually represent the function, so we can confirm the vertex and points like \((0,-\frac{17}{16})\) are correct. This visual check:

• Shows if the parabola really passes through the expected points.
• Confirms the vertex position.

By inputting our function, we can observe the graph's shape and verify all calculated elements. If inconsistencies appear, it's often a sign that you need to recheck calculations or expressions, ensuring that the graph matches theoretical expectations. Using a graphing utility makes complex graphical analysis straightforward and more tangible for learning and verifying mathematical concepts.