Problem 43 - 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. $$ y=5+(x+3)^{2} $$

Step-by-Step Solution

Answer

Shift the parabola y=x^2 left 3 units, then up 5 units. Vertex is at (-3, 5).

Step 1: Identify the Standard Function

The given function is \(y = 5 + (x + 3)^2\). The standard function we start with here is \(y = x^2\), which is the graph of a parabola opening upwards.

Step 2: Transformation 1 - Horizontal Shift

The expression \(x+3\) inside the square indicates a horizontal shift to the left by 3 units. So, we shift the graph of \(y = x^2\) to the left by 3 units.

Step 3: Transformation 2 - Vertical Shift

The constant term 5 outside the square indicates a vertical shift upwards by 5 units. After shifting to the left by 3 units, we shift the entire graph up by 5 units.

Step 4: Analyze Final Graph

The final graph is \(y = (x+3)^2\) shifted up by 5 units. It retains the shape of a parabola opening upwards, with its vertex now at \((-3, 5)\).

Key Concepts

Parabola Graph

A parabola graph is a type of curve where each point is at an equal distance from a fixed point (the focus) and a fixed straight line (the directrix). For the function \( y = x^2 \), the graph is a simple upward-opening parabola with a vertex at the origin \((0,0)\).
The standard parabola \( y = x^2 \) is symmetric about the y-axis, and the vertex is the lowest point on the graph, called the minimum point.

• The equation of a parabola can often be expressed in the form \( y = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants.
• If \( a > 0 \), the parabola opens upwards; if \( a < 0 \), it opens downwards.

The graph does not change direction when zeros or intercepts are present; rather, it rises or falls away from those points depending on the value and sign of \( a \). Understanding this helps in sketching the graph accurately.

Horizontal Shift

The horizontal shift involves moving the graph of a function left or right along the x-axis. Given an expression of \( y = (x + d)^2 \), it signifies a horizontal shift. Remember:

• If \( d > 0 \), the graph shifts to the left by \( d \) units.
• If \( d < 0 \), the graph shifts to the right by \( |d| \) units.

In our example, \( y = 5 + (x+3)^2 \), the expression \( (x+3) \) translates to a horizontal shift of the graph of \( y = x^2 \) to the left by 3 units.
This means every point on the original graph moves 3 units leftward. It's like re-centering the entire graph slightly to the side, which impacts where the vertex—now \((-3, 0)\)—will be located on the x-axis, before any further transformations.

Vertical Shift

A vertical shift moves a graph up or down along the y-axis. This occurs when a constant is added or subtracted from the function. For the general function \( y = f(x) + c \):

• If \( c > 0 \), the graph shifts upward by \( c \) units.
• If \( c < 0 \), the graph shifts downward by \( |c| \) units.

In the function \( y = 5 + (x+3)^2 \), the constant \( +5 \) suggests a vertical shift upward by 5 units. After applying the horizontal shift, the entire graph then moves up 5 units along the y-axis.
This vertical repositioning changes the location of the vertex—from \((-3, 0)\) after the horizontal shift, to \((-3, 5)\) after the vertical shift is applied. Each point on the graph follows suit, leading to the final positioned graph.

Vertex Form

Vertex form is a way to express the equation of a parabola, which makes it easy to determine its vertex and direction. The formula is generally written as \( y = a(x-h)^2 + k \). Here’s how you can interpret it:

• The vertex of the parabola is given by the point \( (h, k) \).
• The value \( a \) determines the width and direction (upward or downward) of the parabola.

In the scenario of \( y = 5 + (x+3)^2 \), we can rewrite this in vertex form as \( y = (x+3)^2 + 5 \). Here:

• \( h = -3 \), so the vertex is horizontally shifted left to \( x = -3 \).
• \( k = 5 \), so the vertex is vertically shifted up to \( y = 5 \).
• Hence, the vertex is \((-3, 5)\).

Understanding the vertex form is crucial because it provides a clear picture of the parabola's positioning and orientation without having to graph it point-by-point.