Question

Identify the vertex, focus, directrix, and axis of symmetry of the parabola. Describe the transformations of the graph of the standard equation with vertex \((\mathbf{0}, \mathbf{0})\). $$y=(x+3)^2-5$$

Short answer

The vertex of the parabola is at \((-3, -5)\), the focus is at \((-3, -19/4)\), and the directrix is \(y = -21/4\). The axis of symmetry is \(x = -3\). The graph of this parabola is a result of shifting the standard parabola 3 units to the left and 5 units down.

Step-by-step solution

Identify the Vertex

The equation of the parabola is given by \(y = (x+3)^2 -5\). In this form, the vertex of the parabola is at \((-h, -k)\), hence the vertex of this parabola is at \((-3, -5)\).

Calculate the value of p

Since our equation doesn't include a \(4p\) value, we know that \(p = 1/4\).

Identify the Focus

The focus of a parabola \(y = (x-h)^2 + k\) is \((h, k+p)\). Plugging in our values, we find the focus to be \(-3, -5+1/4) = (-3, -19/4)\).

Identify the Directrix

The directrix of a parabola \(y = (x-h)^2 + k\) is \(y = k-p\). Plugging in our values, we find the directrix to be \(y = -5 - 1/4 = -21/4\).

Identify the Axis of Symmetry

The axis of symmetry of a parabola in this form is always \(x = h\), thus our axis of symmetry is \(x = -3\).

Describe the Transformations

Compared to the standard parabola \(y = x^2\), this parabola has been shifted 3 units to the left and 5 units down due to the \(-3\) and \(-5\) in the equation, respectively. Hence, the transformations are a horizontal shift to the left by 3 units and a vertical shift downwards by 5 units.

Key concepts

Vertex of a Parabola

Understanding the vertex of a parabola is crucial for graphing and analyzing quadratic functions. The vertex is the highest or lowest point on the parabola, depending on whether the parabola opens upwards or downwards. In the given equation y = (x+3)^2 - 5, we find the vertex by transforming the quadratic into vertex form, y = a(x-h)^2 + k, where (h, k) is the vertex of the parabola. Here, the vertex is at (-3, -5).

Knowing this piece of information, students can begin to sketch the graph and identify other key features related to the parabola's vertex, such as the axis of symmetry and the optimal value.

Focus of a Parabola

The focus of a parabola is a fixed point that, along with the directrix, defines the parabola's shape. They obey the property that for any point on the parabola, its distance to the focus is equal to its distance to the directrix. For the equation y = (x+3)^2 - 5, the focus can be calculated using the formula (h, k + p), where p is the distance from the vertex to the focus. The focus, in our case, is determined to be (-3, -19/4).

The focus, being a specific characteristic of the parabola, is paramount for its proper graphical representation and understanding its reflective properties. Students should remember that the side of the vertex opposite to the focus is where the parabola opens towards.

Directrix of a Parabola

A parabola's directrix is a line that, together with the focus, defines the curve. The directrix is perpendicular to the axis of symmetry and lies outside the parabola itself. For the given equation y = (x+3)^2 - 5, the directrix is given by the formula y = k - p. With our known values, the directrix is y = -21/4.

Understanding the directrix helps students to appreciate the geometric nature of a parabola and its applications in various fields, such as optics and satellite dishes, due to its reflective properties.

Axis of Symmetry

The axis of symmetry is a vertical line that divides the parabola into two mirror images. It passes through the vertex and is thus vital for graphing and understanding the parabola's symmetry. For our function y = (x+3)^2 - 5, the axis of symmetry is at x = -3.

By locating the axis of symmetry, students can easily identify the parabola's reflective property and use it to graph the parabola accurately, being sure that every point on one side has a corresponding point on the other side at equal distances from the axis.

Graph Transformations

Graph transformations are alterations that move or change the shape of a graph. For the parabola given by y = (x+3)^2 - 5, we observe two transformations when compared to the parent function y = x^2: a horizontal shift to the left by 3 units, and a vertical shift down by 5 units. This tells us the parabola has been translated in the Cartesian plane without altering its shape.

Breaking down transformations helps students predict and understand the graphical outcome of alterations applied to the standard form of a function. It's a cornerstone for graphing, as it allows for the manipulation of the graph's position and orientation while preserving its geometric characteristics.