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Chapter 2: Problem 65

In Exercises \(55-66\), find the quadratic function that has the given vertex and goes through the given point. $$\text { vertex: }(2.5,-3.5) \text { point }(4.5,1.5)$$

Short Answer

Expert verified
The quadratic function is \( f(x) = \frac{5}{4}(x-2.5)^2 - 3.5 \).

Step by step solution

01

Understanding Quadratic Functions in Vertex Form

A quadratic function can be written in vertex form as \( f(x) = a(x-h)^2 + k \) where \((h, k)\) is the vertex of the parabola. Here, \((h, k) = (2.5, -3.5)\). So, the function is \( f(x) = a(x-2.5)^2 - 3.5 \).
02

Substitute the Given Point to Find \(a\)

We know the point \((4.5, 1.5)\) lies on the function. Substitute \(x = 4.5\) and \(f(x) = 1.5\) into the equation: \[ 1.5 = a(4.5 - 2.5)^2 - 3.5 \] This simplifies to \(1.5 = a(2)^2 - 3.5 \).
03

Solve for \(a\)

Simplify the expression from step 2: \[ 1.5 = 4a - 3.5 \] Add \(3.5\) to both sides to isolate the term with \(a\): \[ 5 = 4a \] Divide both sides by \(4\) to solve for \(a\):\[ a = \frac{5}{4} \].
04

Write the Final Quadratic Function

Now that we have \(a = \frac{5}{4}\), substitute it back into the vertex form of the function: \[ f(x) = \frac{5}{4}(x-2.5)^2 - 3.5 \]. This is the quadratic function with vertex \((2.5, -3.5)\) that passes through the point \((4.5, 1.5)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Vertex Form
The vertex form of a quadratic function makes it especially easy to identify important characteristics of a parabola, such as its vertex. This particular form is given by the formula:
  • \( f(x) = a(x-h)^2 + k \)
  • Here, \((h, k)\) represents the vertex of the parabola.
The advantage of using vertex form is that it allows us to directly see and use the vertex of the parabola in our calculations.
In our exercise, the vertex given was \((2.5, -3.5)\), and plugging these into \(h\) and \(k\) provides the initial setup for the quadratic function:
  • \( f(x) = a(x-2.5)^2 - 3.5 \)
This representation is straightforward and 鈥媏fficient for finding a particular quadratic function's curve, especially when you know the vertex.
Parabola
A parabola is a U-shaped curve that can open upward or downward. It represents the graphical shape of a quadratic function.
All parabolas have certain common properties, such as:
  • One vertex, which is the point where the parabola changes direction.
  • A central axis of symmetry, which is a vertical line that passes through the vertex of the parabola.
  • The direction of the parabola, determined by the coefficient \(a\) in the vertex form equation. If \(a > 0\), the parabola opens upward, and if \(a < 0\), it opens downward.
The location and width of the parabola's opening are influenced by the values of \(h\), \(k\), and \(a\). By understanding these aspects, one can visualize and create the shape of the parabola on a graph with just a few parameters.
Solving Quadratic Equations
To find a quadratic function that fits given conditions, we often need to solve quadratic equations. This task involves finding the value of \(a\) when a point on the parabola is known along with its vertex.
For the exercise, we had a point \((4.5, 1.5)\). To use this point, it was substituted into the equation:
  • \( 1.5 = a(4.5-2.5)^2 - 3.5 \)
Solving these equations usually involves:
  • Substitute the known \(x\) and \(f(x)\) values.
  • Simplify the equation to isolate the variable \(a\).
  • Solve for \(a\) to find the specific quadratic function.
In this instance, these steps resulted in finding \(a = \frac{5}{4}\).
This value of \(a\) is then used to complete the quadratic function in vertex form, specific to the given points.
Vertex of a Parabola
The vertex of a parabola is a significant feature, marking the maximum or minimum of the function depending on the parabola's orientation.
It is positioned at \((h, k)\) in the vertex form of a quadratic equation:
  • \(f(x) = a(x-h)^2 + k \)
Understanding the vertex helps determine:
  • The parabola's highest or lowest point.
  • How the rest of the parabola moves away symmetrically from this central point.
  • Provides insight into the function's domain and range based on directional opening.
In the example from the exercise, the vertex \((2.5, -3.5)\) is not only the point where the parabola changes direction, but also played a crucial role in setting up and finding the complete quadratic equation.

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