Problem 25 - Algebra and Trigonometry

Question

Use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola's axis of symmetry. Use the graph to determine the function's domain and range. $$ f(x)=4-(x-1)^{2} $$

Step-by-Step Solution

Answer

The vertex of the parabola is at (1, 4), the y-intercept is at (0, 3) and the axis of symmetry is \(x = 1\). The parabola opens downwards. The domain of the function is all real numbers and the range is \(-\infty < y \leq 4\).

Step 1: Finding the Vertex

The vertex form of a parabola's equation is usually expressed as \(f(x) = a(x-h)^2 + k\), where (h, k) are the coordinates of the vertex. So in the provided equation, the vertex of the parabola is (1, 4) .

Step 2: Finding the Y-intercept

The y-intercept of a function is the value of f(x) when x = 0. So, if we replace x with 0 in the given function, we get \(f(0)= 4-(0-1)^{2} = 3\) . So the y-intercept is (0, 3).

Step 3: Finding the Axis of Symmetry

The axis of symmetry for any parabola is given by the x-value of the vertex. In this case, the x-coordinate of the vertex is 1, therefore, the equation of the axis of symmetry is \(x = 1\).

Step 4: Sketching the Graph

Plot the vertex point (1, 4), the y-intercept (0, 3) and the axis of symmetry \(x = 1\). Then sketch the parabola, which opens downwards because the coefficient of \(x^{2}\) is negative.

Step 5: Determining the Domain and Range

The domain of a quadratic function is all real numbers because a parabola extends indefinitely to the left and right. So, the domain is \(-\infty < x < \infty\). The range of a function is the set of possible y values. Since the parabola opens downwards, the highest point or the maximum y value is at the vertex. So, the range is \(-\infty < y \leq 4\).

Key Concepts

Vertex Form

In quadratic equations, converting to vertex form can be very helpful. Vertex form is a particular way of writing quadratic functions and it looks like this: \[ f(x) = a(x-h)^2 + k \] Here,

• \(a\) does not equal zero and is the coefficient that affects the "width" or "narrowness" of the parabola and its opening direction.
• \(h\) and \(k\) are crucial because they give us the exact coordinate of the vertex, \((h, k)\).

The vertex of our given function \( f(x)=4-(x-1)^{2} \) is \((1, 4)\). This is because the equation matches the vertex form, where \(h = 1\) and \(k = 4\). Knowing the vertex helps not only in sketching graphs but also in understanding the symmetry and the maximum or minimum values of a quadratic function.
To convert a standard form into vertex form, you often need to complete the square, which is a method of algebraically manipulating an equation.

Axis of Symmetry

Every parabola has an axis of symmetry. This is an imaginary line that divides the parabola into two mirror-image halves. For any parabola in vertex form, its axis of symmetry can easily be determined by looking at the x-coordinate of the vertex, \(x = h\).
In our function \(f(x) = 4-(x-1)^2\), the axis of symmetry is the line \(x = 1\). Understanding this helps in:

• Precisely drawing the graph,
• Predicting how the parabola behaves,
• Aiding in finding other points on the graph due to symmetry.

Imagine folding the graph along this line; the two sides would perfectly align with each other. The concept of symmetry is integral as it simplifies many calculations and interpretations when dealing with quadratics.

Parabola

A parabola is the shape you see when you graph a quadratic function. This U-shaped curve can open upwards or downwards:

• If \(a > 0\), it opens upwards.
• If \(a < 0\), it opens downwards.

In the case of \(f(x) = 4-(x-1)^2\), since the coefficient of \( (x-1)^2 \) is negative, it is clear that the parabola opens downwards. This indicates that the parabola has a maximum point at its vertex, which is \((1, 4)\).
Recognizing whether a function represents a parabola and its orientation helps predict behaviors like where it might intersect the x-axis (roots) and how it extends across the coordinate plane.

Domain and Range

The domain and range are essential aspects of any function. For quadratic functions, the domain is crystal clear because the parabola continues indefinitely left and right, accepting all real x values:\[-\infty < x < \infty\]
The range, on the other hand, depends on the vertex, and the direction in which the parabola opens. Since our parabola opens downwards and reaches a maximum at \((1, 4)\), the range consists of all y values that are less than or equal to 4. Thus, the range is given by:\(-\infty < y \leq 4\)
This means that, although x can take on any possible value, y is restricted to values that reflect the downward opening of the parabola, reaching its peak at the vertex.