Problem 44 - Algebra and Trigonometry

Question

$$ \begin{array}{l}{\text { Find a function whose graph is a parabola with vertex }(3,4)} \\ {\text { and that passes through the point }(1,-8) .}\end{array} $$

Step-by-Step Solution

Answer

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

Step 1: Understand the Parabola's Vertex Form

The vertex form of a parabola is given by \( y = a(x-h)^2 + k \), where \((h, k)\) is the vertex of the parabola. In this problem, the vertex is \((3,4)\). Thus, the equation becomes \( y = a(x-3)^2 + 4 \).

Step 2: Use the Given Point to Find 'a'

The parabola passes through the point \((1, -8)\). So, substitute \(x = 1\) and \(y = -8\) into the equation \( y = a(x-3)^2 + 4 \). This gives \(-8 = a(1-3)^2 + 4\).

Step 3: Simplify and Solve for 'a'

Evaluate the equation \(-8 = a(1-3)^2 + 4\). This simplifies to \(-8 = a(4) + 4\) \([since \;(1-3)^2 = 4]\). Subtract 4 from both sides to get \(-12 = 4a\), then divide both sides by 4 to find \(a\): \(a = -3\).

Step 4: Write the Final Function

Substitute \(a = -3\) back into the vertex form equation. The function is \( y = -3(x-3)^2 + 4 \). This is the equation of the parabola with the given vertex and passing through the specified point.

Key Concepts

Parabola Equation

A parabola is a U-shaped curve that can open upwards or downwards. Its general equation can be expressed in two main forms: the standard form and the vertex form. The vertex form is particularly helpful when you need to identify specific characteristics of a parabola, such as its vertex. The vertex form of a parabola equation is: \\[ y = a(x - h)^2 + k \] \Here, \(a\) is a coefficient that affects the width and direction of the parabola. The point \((h, k)\) represents the vertex of the parabola. This makes it straightforward to see where the turning point or the peak/lowest point is located on the graph. \

\
• \(a > 0\): parabola opens upwards
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• \(a < 0\): parabola opens downwards
\

\Knowing the vertex form of a parabola helps you understand its shape and direction, which are essential when graphing or analyzing conic sections.

Vertex of a Parabola

The vertex of a parabola is a significant point that represents the highest or lowest point on a parabola depending on its direction. This point is crucial for graphing and provides a lot of information about the parabola's orientation. \In the vertex form of a parabola equation, \((h, k)\) is the vertex. If the equation is \\[ y = a(x - h)^2 + k \], \then the vertex is simply \((h, k)\). \- If \(a > 0\), the vertex is the lowest point (a minimum). \- If \(a < 0\), the vertex is the highest point (a maximum). \Finding the vertex is easy when you have the equation in vertex form, which makes this form particularly useful for solving problems related to parabolas. Understanding where the vertex lies allows you to better graph and conceptualize the parabola.

Solving for Coefficient

In solving a parabola problem, determining the coefficient \(a\) is important for understanding how the parabola stretches or compresses and in which direction it opens. When a parabola passes through a known point, you can use this information to solve for the coefficient \(a\). \Given the equation in vertex form, \\[ y = a(x - h)^2 + k \], \substitute the coordinates of the known point into the equation for \(x\) and \(y\). From there, simplify and solve the equation to find \(a\). \For example, if the vertex is \((3, 4)\) and a point on the parabola is \((1, -8)\), substitute \(x = 1\) and \(y = -8\) to find \(-8 = a(1-3)^2 + 4\). \After simplifying and solving, you find that \(a = -3\). This process allows you to write the complete vertex form equation of the parabola, incorporating both its vertex and the coefficient that defines its shape. Knowing \(a\) gives insights into the parabola's steepness and direction.