Problem 44 - Precalculus Mathematics for Calculus

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 \(f(x) = -3(x-3)^2 + 4\).

Step 1: Understanding the Structure of a Parabola

A parabola with vertex \((h, k)\) \ can \ be \ expressed \ in \ the \ vertex \ form \ as \\[\ f(x) = a(x-h)^2 + k\]\. \Here, \ (h,k) \= (3,4).\

Step 2: Substitute the Vertex into the Equation

Substitute the vertex \((3,4)\)\ into \ the \ general \ formula \ to \ get: \\[\ f(x) = a(x-3)^2 + 4.\]},

Step 3: Use the Point the Parabola Passes Through

We know the parabola passes through \((1,-8)\)\ , which means \\[\ f(1) = -8.\]\. \ Substituting \ the \ point \ into \ the \ equation \ we \ have: \\[\ -8 = a(1-3)^2 + 4.\]\.\\ \ Simplify: \\[\ -8 = a(4) + 4.\]

Step 4: Solve for the Coefficient a

Rearrange the equation to solve for \(a\)\:\[\ -8 - 4 = 4a\]\ \ \\[\ -12 = 4a\] \ \\[\ a = -3.\]

Step 5: Write the Final Equation of the Parabola

Substitute \(a = -3\)\ back\ into \ the \ equation \ we \ derived \ earlier: \\[\ f(x) = -3(x-3)^2 + 4\]}],

Step 6: Short Answer

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

Key Concepts

vertex form

The vertex form of a quadratic function makes understanding the properties of a parabola straightforward. It's written as \( f(x) = a(x-h)^2 + k \), where \((h, k)\) represents the vertex of the parabola. This form is particularly useful because you can easily identify the vertex of the parabola directly from the equation, which is a big advantage.
When looking at the vertex form:

• \( a \) is a coefficient that affects the direction and width of the parabola. If \( a > 0 \), the parabola opens upwards, while if \( a < 0 \), it opens downwards.
• \( (h, k) \) is the vertex, giving you the highest or lowest point of the parabola depending on the direction it opens.

By substituting the vertex coordinates into the vertex form, you can easily construct the basis of your quadratic equation. This method simplifies many problems, as it doesn't require complex factoring to find the vertex like in standard forms.

quadratic function

Quadratic functions are essential in algebra and graphing due to their parabolic shape. The general form of a quadratic function is \( ax^2 + bx + c \), but the vertex form \( f(x) = a(x-h)^2 + k \) offers clear insights into graph features.
Key aspects of quadratic functions include:

• The graph is always a parabola, which may open up or down.
• Quadratic functions always include three coefficients \( a, b, \ and \ c \) in their standard form, dictating the parabola's shape and direction.
• The solutions or roots of the quadratic equation \( ax^2 + bx + c = 0 \) correspond to the "x" values where the parabola intersects the x-axis.

Understanding these properties helps not just in graphing, but also in applications involving maximum and minimum values, which can be directly retrieved from the vertex in vertex form.

vertex of the parabola

The vertex of the parabola is a crucial concept in understanding quadratic functions and their applications. It represents the turning point of the curve. In the vertex form \( f(x) = a(x-h)^2 + k \), the vertex is easily identified as \((h, k)\).
Special features of the parabola's vertex include:

• If the parabola opens upward (a smiley \(:)\), the vertex is the lowest point.
• If the parabola opens downward (a frowny \(:()\), the vertex is the highest point.
• The vertex is a point of symmetry, meaning the parabola is mirrored along the vertical line through the vertex.

Identifying the vertex quickly allows for efficient graphing and problem-solving, as it provides a starting point for sketching the whole function and for calculating optimization tasks, especially in physics and engineering problems.