Question

\(g(x)=2(x+1)^2-3\)

Short answer

The function \(g(x) = 2(x+1)^2 - 3\) corresponds to a parabola with vertex (-1, -3), opening upwards and narrower due to the value 2 of parameter \(a\).

Step-by-step solution

Identify the form

Firstly, let's identify the function form. Here we have a quadratic function in a specific form called the vertex form. The vertex form is \(f(x) = a(x-h)^2 + k\), where \(h\) and \(k\) are the coordinates of the vertex of the graph of this function. The vertex form allows us to easily identify the vertex of the parabola by looking at \(h\) and \(k\), and the value of \(a\) indicates if the parabola opens upwards or downwards (if \(a\) is positive the parabola opens upwards, otherwise it opens downwards) and how steep it is (the larger the absolute value of \(a\), the steeper the parabola).

Identify the vertex

The vertex of this parabola is given by \((-h, k)\), where \(h\) and \(k\) come from the form of our function \(f(x) = a(x-h)^2 + k\). This results in the vertex of the function being at (-1, -3).

Identify the direction

The direction of the parabola is given by the sign of the parameter \(a\). In our function, \(a = 2\), which is greater than zero. This means our parabola opens upwards.

Identify the width

The width of our parabola is determined by the absolute value of \(a\), which in this case is 2. The larger the absolute value of \(a\), the narrower the parabola. So, in our case, the parabola is narrower than if \(a\) were 1.

Key concepts

Understanding Quadratic Functions

A quadratic function is a type of function that can be defined by an equation of the form \( f(x) = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants and \( a eq 0 \). Quadratic functions are fundamental in mathematics because they describe a wide range of natural phenomena, from the trajectory of a flying object to the profit calculations in business.

The general shape of the graph of a quadratic function is known as a parabola, which is a U-shaped curve that can open either upward or downward. This shape makes quadratic functions vital for disciplines like physics and engineering. The graph's direction and width depend on the coefficient \( a \). Specifically, if \( a > 0 \), the parabola opens upwards, while if \( a < 0 \), it opens downwards.

• The vertex of a parabola represents its turning point, the point where the graph changes direction.
• The axis of symmetry is a vertical line that passes through the vertex and divides the parabola into two mirror images.
• The leading coefficient \( a \) affects the steepness or narrowness of the parabola.

By understanding these characteristics, anyone can analyze and predict the behavior of quadratic functions in different scenarios.

Exploring Parabolas

Parabolas are intriguing curves that are symmetrical and can be seen frequently in the real world. Each quadratic function plots a parabola on a graph. What makes parabolas interesting is their unique properties and how they are influenced by their equation's coefficients.

The vertex of the parabola is a crucial element. It is the highest or lowest point on the graph, depending on the parabola's orientation. In the vertex form of a quadratic function, \( f(x) = a(x-h)^2 + k \), the vertex is the point \((h, k)\). This makes it particularly simple to identify the vertex just by looking at the equation.

Parabolas also have an axis of symmetry, which is a line that runs through the vertex and neatly splits the parabola into two identical halves. This axis helps in reflecting the symmetrical nature of a parabola.

• When \( a \) is positive, the parabola opens upwards, forming a U shape.
• When \( a \) is negative, it opens downwards, forming an upside-down U shape.
• The absolute value of \( a \) determines how "wide" or "narrow" the parabola appears on the graph. A larger value results in a narrower parabola.

These properties allow us to effectively analyze and draw parabolas for any given quadratic function.

Graphing Quadratic Functions

Graphing quadratic functions provides a visual representation of the function's behavior. With the quadratic function in vertex form \( g(x) = 2(x+1)^2 - 3 \), we can graph by identifying key components such as the vertex, axis of symmetry, and direction in which the parabola opens.

To graph \( g(x) \), first identify the vertex from the equation. Here, the vertex is at \((-1, -3)\). This is obtained from \((h, k)\) in the equation \( a(x-h)^2 + k \). Next, determine the parabola's direction by inspecting the sign of \( a \). Since \( a = 2 \) is positive, the parabola opens upwards, creating a U shape.

The axis of symmetry can be found using the equation of the vertex. It's a vertical line \( x = -1 \), passing through the vertex. This line helps in drawing the graph accurately by providing a central guideline.

• Step 1: Plot the vertex \((-1, -3)\) on the graph.
• Step 2: Sketch the axis of symmetry line, \( x = -1 \).
• Step 3: Choose a few values for \( x \) on either side of the axis of symmetry, calculate corresponding \( g(x) \) values, and plot them.
• Step 4: Connect the points with a smooth curve to complete the parabola.

Positioning the vertex and drawing the symmetry line helps ensure the parabola is drawn accurately, thus vividly portraying the quadratic function.