Question

On one set of coordinate axes, graph the family of parabolas \(f(x)=x^{2}+b x+1\) for \(b=-4, \quad b=0,\) and \(b=4 .\) Describe the general characteristics of this family.

Short answer

The family has parabolas that all open upwards with vertices and axes of symmetry varying by \(b\) values.

Step-by-step solution

Identify the given function

The given family of parabolas is represented by the equation: \(f(x)=x^2 + bx + 1\)

Substitute values of \(b\)

Substitute each of the given values of \(b\) into the equation to obtain specific parabolas: - For \(b = -4\): \(f(x) = x^2 - 4x + 1\)- For \(b = 0\): \(f(x) = x^2 + 1\)- For \(b = 4\): \(f(x) = x^2 + 4x + 1\)

Find the vertex of each parabola

Use the formula to find the vertex of a parabola represented by \(g(x) = ax^2 + bx + c\):The vertex is at \(x = -\frac{b}{2a}\). For each equation:- For \(b = -4\): \(x = \frac{4}{2} = 2\), thus the vertex is at \((2, f(2)) = (2, -3)\)- For \(b = 0\): \(x = 0\), thus the vertex is at \((0, f(0)) = (0, 1)\)- For \(b = 4\): \(x = -\frac{4}{2} = -2\), thus the vertex is at \((-2, f(-2)) = (-2, -3)\)

Determine the axis of symmetry

The axis of symmetry for a parabola given by \(g(x) = ax^2 + bx + c\) is \(x = -\frac{b}{2a}\). Use the same calculations as in Step 3:- For \(b = -4\): Axis is \(x = 2\)- For \(b = 0\): Axis is \(x = 0\)- For \(b = 4\): Axis is \(x = -2\)

Plot the parabolas

Plot each parabola on the same set of coordinate axes:- \(f(x) = x^2 - 4x + 1\) with vertex (2, -3) and axis of symmetry \(x = 2\)- \(f(x) = x^2 + 1\) with vertex (0, 1) and axis of symmetry \(x = 0\)- \(f(x) = x^2 + 4x + 1\) with vertex (-2, -3) and axis of symmetry \(x = -2\)

Describe the general characteristics

The family of parabolas \(f(x) = x^2 + bx + 1\) has the following general characteristics:- All parabolas open upwards since the coefficient of \(x^2\) is positive.- The vertex location changes based on the value of \(b\).- The parabolas are symmetric around their respective axes of symmetry.

Key concepts

Parabolas

A parabola is a symmetrical, curved shape that is graphed as the equation of a quadratic function. In our given exercise, the family of parabolas is represented by the equation: \( f(x) = x^2 + bx + 1 \).When graphing parabolas:

• The general form of a quadratic function is \( f(x) = ax^2 + bx + c \).
• The direction in which the parabola opens depends on the coefficient of \( x^2 \). If positive, it opens upwards. If negative, it opens downwards.
• Each parabola is symmetric and has a specific axis of symmetry and vertex.

In our given equations, the coefficient of \( x^2 \) is always positive (\( a = 1 \)), thus all the parabolas open upwards.

Vertex

The vertex of a parabola is the highest or lowest point, depending on the direction the parabola opens. In mathematical terms, for a parabola given by \( f(x) = ax^2 + bx + c \), the vertex can be found using the formula \( x_{vertex} = -\frac{b}{2a} \). For our parabolas:

• For \( b = -4 \): The vertex is at \( x = \frac{4}{2} = 2 \). Plugging \( x = 2 \) back into the equation gives the vertex at (2, -3).
• For \( b = 0 \): The vertex is at \( x = 0 \). Plugging \( x = 0 \) into the equation gives the vertex at (0, 1).
• For \( b = 4 \): The vertex is at \( x = -\frac{4}{2} = -2 \). Plugging \( x = -2 \) into the equation gives the vertex at (-2, -3).

The vertex provides vital information for graphing, as it gives the turning point of the parabola.

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. This line can always be found using the same \( x \) value as the vertex.

• For \( b = -4 \): Axis of symmetry is \( x = 2 \).
• For \( b = 0 \): Axis of symmetry is \( x = 0 \).
• For \( b = 4 \): Axis of symmetry is \( x = -2 \).

Understanding the axis of symmetry is crucial for graphing because it helps to ensure that the parabola is plotted accurately and symmetrically.

Graphing

Graphing quadratic functions involves plotting the parabolas on a coordinate plane:

• Start by identifying the vertex and the axis of symmetry for each parabola.
• Use additional points to define the shape. For example, substituting values of \( x \) to find corresponding \( y \) values.
• For \( b = -4 \): Plot the parabola \( f(x) = x^2 - 4x + 1 \) with vertex (2, -3) and axis of symmetry \( x = 2 \).
• For \( b = 0 \): Plot the parabola \( f(x) = x^2 + 1 \) with vertex (0, 1) and axis of symmetry \( x = 0 \).
• For \( b = 4 \): Plot the parabola \( f(x) = x^2 + 4x + 1 \) with vertex (-2, -3) and axis of symmetry \( x = -2 \).

In graphing, always remember that each point is a reflection across the axis of symmetry. Observing the vertex and symmetry helps in sketching the parabolas accurately.