Problem 35 - Algebra and Trigonometry

Question

33–48 ? Sketch the graph of the function, not by plotting points, but by starting with the graph of a standard function and applying transformations. $$ f(x)=-(x+1)^{2} $$

Step-by-Step Solution

Answer

The graph is a downward opening parabola with vertex at (-1, 0).

Step 1: Identify the Parent Function

The given function is a transformation of the quadratic function. The parent function is the standard quadratic function: \[ y = x^2 \]

Step 2: Apply Horizontal Shift

The function \( f(x) = -(x+1)^2 \) includes a transformation inside the parentheses. The \(+1\) inside the parentheses indicates a horizontal shift to the left by 1 unit. The new function is:\[ y = x^2 \rightarrow y = (x+1)^2 \] This shifts the vertex of the parabola from the origin to \((-1, 0)\).

Step 3: Apply Reflection

In the function \( f(x) = -(x+1)^2 \), the negative sign outside the squared term is a vertical reflection over the x-axis. This flips the graph upside down, so:\[ y = (x+1)^2 \rightarrow y = -(x+1)^2 \] Now, the parabola opens downward, with its vertex remaining at \((-1, 0)\).

Step 4: Graph the Transformed Function

Using the transformations identified, sketch the graph. Start with the vertex at \((-1, 0)\). Since the parabola opens downward (due to the reflection), draw it extending downwards symmetrically from the vertex.

Key Concepts

Quadratic Function

A quadratic function is a type of polynomial where the highest degree of the variable is two. The standard form of a quadratic function is given by:

• \[ y = ax^2 + bx + c \]
• where \(a\), \(b\), and \(c\) are constants.
• The coefficient \(a\) determines the direction of the parabola (upwards or downwards).

Quadratic functions are known for their characteristic U-shaped graphs called parabolas. These functions can model various real-world scenarios, such as projectile motion.
Understanding transformations of quadratic functions is crucial because they allow us to predict and analyze changes to the graph based on modifications to the equation.

Horizontal Shift

A horizontal shift occurs when a function is moved left or right along the x-axis. This shift is part of the broader category of transformations and helps in adjusting the position of a graph.
In the case of quadratic functions, consider the transformation from \( y = x^2 \) to \( y = (x+1)^2 \).

• The \(+1\) inside the parentheses indicates that each point on the graph is shifted 1 unit to the left.
• The vertex, which is the highest or lowest point of a parabola, moves from \((0, 0)\) to \((-1, 0)\).

Horizontal shifts do not change the shape of the graph, just its position on the x-axis.

Vertical Reflection

A vertical reflection is a transformation that flips a graph over a line, typically the x-axis. The reflection changes the direction in which the parabola opens.
For the function \( f(x) = -(x+1)^2 \):

• The negative sign outside the squared term causes the graph\'s reflection over the x-axis.
• This means if the original parabola opens upwards, after reflection, it will open downwards.

The vertex remains in the same horizontal position after reflection, but the parabola changes its orientation. Vertical reflection is key in determining the direction of the opening of quadratic graphs.

Parabola

A parabola is the graphical representation of a quadratic function. It is a symmetrical and curved shape that opens either upwards or downwards.

• The vertex is the central point of a parabola, indicating its highest or lowest value depending on the direction it opens.
• The axis of symmetry is the vertical line that passes through the vertex, dividing the parabola into two mirror image halves.
• The direction in which the parabola opens is determined by the sign of the coefficient in front of the \(x^2\) term.

Parabolas are important in mathematics due to their properties and their appearance in various contexts, such as physics, engineering, and computer graphics.