Question

\(f(x)=-4(x+1)^2-5\)

Short answer

The vertex of the function \(f(x)=-4(x+1)^2-5\) is \((-1,-5)\). The parabola of the function opens downwards and the axis of symmetry is \(x=-1\).

Step-by-step solution

Identify the vertex form

A quadratic function in vertex form is expressed as \(f(x)=a(x-h)^2+k\), where \((h, k)\) are the coordinates of the vertex of the parabola. The given function is \(f(x)=-4(x+1)^2-5\), so by comparison, \(a=-4\), \(h=-1\), and \(k=-5\).

Identify the Vertex

From Step 1, identified that the vertex coordinates \((h, k)\) are \((-1, -5)\). So the vertex of the given quadratic function is \((-1, -5)\).

Identify the direction of the parabola

The value of \(a\) in the vertex form determines the direction of the parabola. If \(a>0\), the parabola opens upwards, if \(a<0\), it opens downwards. Here, \(a=-4\), which is less than 0. So, the parabola opens downwards.

Identify the axis of symmetry

The axis of symmetry of a parabola given in vertex form is the line \(x=h\). In this case, since \(h=-1\), the axis of symmetry is \(x=-1\).

Key concepts

Vertex Form

The vertex form of a quadratic function is a powerful way to understand the behavior of parabolas. This is written as \( f(x)=a(x-h)^2+k \) where \( (h, k) \) represents the vertex of the parabola. This form makes it easy to determine key properties of the quadratic function. In the given example, \( f(x) = -4(x+1)^2-5 \), you can see how the function fits neatly into the vertex form:

• \( a = -4 \)
• \( h = -1 \)
• \( k = -5 \)

Therefore, the vertex of this particular function is at the point \((-1, -5)\). Knowing the vertex helps you immediately pinpoint the lowest or highest point on the graph, depending on the parabola's direction.

Parabola Direction

The direction in which a parabola opens is determined by the value of \( a \) in the vertex form. This is a simple rule:

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

For the quadratic function \( f(x) = -4(x+1)^2 - 5 \), the value of \( a \) is -4. Since this is less than zero, the parabola opens downwards. This means the vertex represents the highest point on the graph. Understanding the direction helps when sketching the function and predicting its behavior.

Axis of Symmetry

The axis of symmetry of a parabola is an important feature that gives the line which vertically divides the parabola into two mirror images. For a quadratic equation in vertex form \( f(x) = a(x-h)^2 + k \), the axis of symmetry is the line \( x = h \).
For \( f(x) = -4(x+1)^2 -5 \), the axis of symmetry can be easily found. Since \( h = -1 \), the axis of symmetry is the line \( x = -1 \). This simply means that the parabola is symmetrical about this vertical line, and knowing it helps when graphing the parabola. Any point on the graph has a corresponding point directly across this line.

Graphing Quadratics

Graphing quadratic functions using the vertex form is straightforward once you understand their key components. To graph the parabola for \( f(x) = -4(x+1)^2 -5 \):

• First, plot the vertex, which is at \((-1, -5)\). This point is a crucial starting point.
• Next, draw the axis of symmetry, which is the line \( x = -1 \).
• Since \( a = -4 \), note that the parabola opens downwards.
• Choose additional x-values around the vertex to find corresponding y-values, which provides more points to help shape the curve of the parabola.

By keeping these steps in mind, you can visualize and sketch the graph effectively. Pay attention to the width of the parabola as well, as the absolute value of \( a \) (4 in this case) indicates how "stretched" or "compressed" the graph is compared to a standard parabola. A larger absolute value of \( a \) implies a steeper graph.