Problem 35 - Algebra A Combined Function

Question

Graph each quadratic function. Label the vertex and sketch and label the axis of symmetry. $$ f(x)=-(x-1)^{2} $$

Step-by-Step Solution

Answer

Vertex: (1, 0); Axis of symmetry: x = 1; Opens downwards.

Step 1: Identify the Form of the Quadratic

The given quadratic function is \( f(x) = -(x-1)^2 \). This is in the vertex form of a quadratic equation, \( y = a(x-h)^2 + k \), where \( a = -1 \), \( h = 1 \), and \( k = 0 \).

Step 2: Determine the Vertex

Using the vertex form of the quadratic \( f(x) = -(x-1)^2 \), the vertex is at the point \( (h, k) \). Here, \( h = 1 \) and \( k = 0 \), thus the vertex is \( (1, 0) \).

Step 3: Identify the Axis of Symmetry

The axis of symmetry for a quadratic in vertex form \( y = a(x-h)^2 + k \) is the vertical line \( x = h \). Here, \( x = 1 \) is the axis of symmetry.

Step 4: Graph the Function

Sketch the quadratic by plotting the vertex at \( (1, 0) \). Since the coefficient \( a = -1 \) (which is negative), the parabola opens downwards. Draw the axis of symmetry as a dashed line through \( x = 1 \).

Key Concepts

Vertex of a Quadratic Function

The vertex of a quadratic function is a crucial point that represents either the maximum or minimum value of the quadratic equation. For the quadratic function in the exercise, which is given by \[ f(x) = -(x-1)^2 \], the vertex plays a key role.In vertex form, the vertex is at the point \( (h, k) \).

• Here, we see \( h = 1 \) and \( k = 0 \).
• This means the vertex of this function is at the point \( (1, 0) \).

The vertex, in this case, represents the highest point on the graph since the parabola opens downward.
This occurs because the coefficient \( a = -1 \) is negative. The vertex gives the turning point of the parabola and is a useful point for sketching the graph.

Axis of Symmetry

The axis of symmetry is a vertical line that runs through the vertex of the parabola, dividing it into two mirror-image halves. It is a significant part of graphing quadratic functions because it helps in understanding the symmetry and shape of the parabola.
For the function \( f(x) = -(x-1)^2 \), we can identify the axis of symmetry directly from the equation.

• The formula tells us that the axis of symmetry is given by \( x = h \).
• In this case, \( h = 1 \), so our axis of symmetry is \( x = 1 \).

When you draw the graph, you can sketch a dashed line at \( x = 1 \) to represent this axis.
It helps visually in plotting the mirror image of points on one side of the parabola to the other.

Vertex Form of a Quadratic Equation

Understanding the vertex form of a quadratic equation gives us insight into the essential aspects of graphing these functions.
A quadratic function is expressed in vertex form as: \[ y = a(x-h)^2 + k \].

• Here, \( a \) determines the direction of the parabola (upward if positive, downward if negative).
• \( h \) and \( k \) are constants that allow us to locate the vertex \((h, k)\).

For our function \( f(x) = -(x-1)^2 \), we have it already in vertex form: \( a = -1 \), \( h = 1 \), and \( k = 0 \).

• The negative \( a \) tells us the parabola opens downwards.
• The vertex at \( (1, 0) \) makes this the highest point on the graph.

Recognizing and understanding this form can simplify the process of graphing and analyzing quadratic functions.