Problem 40 - Algebra and Trigonometry

Question

Find a function whose graph is a parabola with vertex \((3,4)\) and that passes through the point \((1,-8) .\)

Step-by-Step Solution

Answer

The function is \( y = -3(x-3)^2 + 4 \).

Step 1: Understand the Parabola Vertex Form

The standard form of a parabola with vertex \( (h,k) \) is given by \((y = a(x-h)^2 + k)\). Here, \(h=3\) and \(k=4\). Thus, the function initially looks like \(y = a(x-3)^2 + 4\).

Step 2: Use the Given Point to Solve for 'a'

We know the parabola passes through \( (1, -8) \). Substitute \(x=1\) and \(y=-8\) into the equation: \\\(-8 = a(1-3)^2 + 4\)\\This simplifies to: \(-8 = 4a + 4\)\

Step 3: Solve for 'a'

Subtract 4 from both sides to get \(-12 = 4a\). Divide both sides by 4 to solve for \(a\): \[a = -3\]

Step 4: Write the Final Equation

Now, substitute \(a = -3\) back into the vertex form equation: \[y = -3(x-3)^2 + 4\] This is the equation of the parabola.

Key Concepts

Quadratic Equation

A quadratic equation is a type of polynomial equation that has a degree of two, typically written as \(ax^2 + bx + c = 0\). This type of equation is called "quadratic" because "quad" means square, indicating the highest exponent of the variable is two. Quadratic equations are fundamental in algebra and have extensive applications, such as in physics to describe the motion of objects or in engineering for structural analysis.
The quadratic equation can be expressed in different forms, one of which is the vertex form. This form is especially useful when dealing with parabolas, as it allows us to easily identify the vertex. The vertex form is given by \(y = a(x-h)^2 + k\), where \((h,k)\) is the vertex of the parabola. Understanding how to manipulate quadratic equations is essential in solving for unknown variables and predicting the behavior of parabolic graphs.

Vertex of a Parabola

The vertex of a parabola is a key point on its graph, representing either the highest or lowest point, depending on whether the parabola opens upwards or downwards. It is crucial because it can provide vital information about the parabola's direction and location. In the vertex form of a parabola \(y = a(x-h)^2 + k\), the vertex is directly given by the point \((h, k)\). This makes the vertex form particularly helpful for graphing purposes and problem-solving where the vertex position is specified.
To determine whether a parabola opens upwards or downwards, we look at the coefficient \(a\) in the vertex form. If \(a\) is positive, the parabola opens upwards, and the vertex is its minimum point. If \(a\) is negative, the parabola opens downwards, and the vertex is its maximum point.

• Vertex upwards: \(a > 0\).
• Vertex downwards: \(a < 0\).

Understanding this concept enables you to visualize and interpret the graph of a parabolic function accurately. For example, in the problem's solution, the vertex \((3, 4)\) is identified, and the parabola opens downwards with \(a = -3\). This visualization can aid in verifying the correctness of the derived equation.

Solving for a Variable

Solving for a variable is an essential skill in algebra, often involving isolating the variable of interest on one side of the equation. In quadratic equations, this typically requires rearranging the equation to determine unknown parameters or values. In the context of the vertex form, knowing a specific point that the parabola passes through is critical for determining the coefficient \(a\).
To solve for \(a\) in the given problem, the known point \((1, -8)\) is substituted into the vertex form equation \(y = a(x-3)^2 + 4\). By doing so, we obtain the equation \(-8 = a(1-3)^2 + 4\). Carrying out the arithmetic operations:

• Calculate the square: \((1-3)^2 = 4\).
• Subtract 4 from both sides: \( -8 - 4 = -12\).
• Divide by 4 to solve for \(a\): \(a = -3\).

After determining \(a\), this value is substituted back to complete the vertex form equation, resulting in \(y = -3(x-3)^2 + 4\). This form not only expresses the quadratic equation with known parameters but also verifies that the function's graph passes through the specified point, ensuring the solution's accuracy.