Problem 49 - Algebra 2

Question

Compare the graphs of \(y=2(x-5)^{2}+4\) and \(y=2(x-4)^{2}-1\)

Step-by-Step Solution

Answer

The graphs are identical in shape, but the first is a translation of the second, moved 1 unit right and 5 units up.

Step 1: Identify the Vertex Forms

The given functions are in vertex form. For a parabola in the form \(y = a(x-h)^2 + k\), the vertex is \((h,k)\). Therefore, these parabolas have the form \(y = 2(x-5)^2 + 4\) and \(y = 2(x-4)^2 - 1\).

Step 2: Determine the Vertices

For the function \(y = 2(x-5)^2 + 4\), the vertex is \((5, 4)\). For the function \(y = 2(x-4)^2 - 1\), the vertex is \((4, -1)\).

Step 3: Compare the Parabola Openings and Widths

Both parabolas have the same coefficient \(a = 2\) in front of the quadratic term, indicating that they open upwards and have the same width. The parabolas are narrower than the standard parabola \(y = (x-h)^2\) because \(a > 1\).

Step 4: Compare the Vertices

The vertex for \(y = 2(x-5)^2 + 4\) is \(1\) unit to the right and \(5\) units above the vertex of \(y = 2(x-4)^2 - 1\). This indicates a horizontal shift to the right and a vertical shift upward for the first equation compared to the second.

Step 5: Compare the Graphs

Since both parabolas have the same shape and orientation, but different vertices, \(y = 2(x-5)^2 + 4\) is the translation of \(y = 2(x-4)^2 - 1\) by moving \(1\) unit to the right and \(5\) units up.

Key Concepts

Parabola

A parabola is a U-shaped curve that can open either upwards or downwards. It is the graph of a quadratic function, which is a function that can be written in the form of a polynomial of degree two.
Parabolas have a specific symmetry around a central line called the axis of symmetry. The lowest or highest point on the parabola, depending on its orientation, is called the vertex.
Parabolas can be translated or shifted on the coordinate plane, which changes their position without altering their shape.

• When a parabola opens upwards, as in our examples, it has a minimum point at the vertex.
• The vertex provides essential information about the parabola's position.

Understanding the vertex will help you predict how the entire parabola will be positioned in the graph.

Quadratic Function

A quadratic function is an algebraic equation that describes a parabola. It generally takes the form: \[ y = ax^2 + bx + c \] where \(a\), \(b\), and \(c\) are constants and \(a eq 0\).
The parabola's direction (up or down) is determined by the coefficient \(a\):

• If \(a\) is positive, the parabola opens upwards.
• If \(a\) is negative, it opens downwards.

The quadratic function can also be expressed in vertex form: \( y = a(x-h)^2 + k \), which directly gives the vertex's coordinates as \((h, k)\).
The advantage of the vertex form is that it allows easy identification and manipulation of the graph’s position through \(h\) and \(k\). Knowing these values makes it simpler to perform translations and shifts on the graph.

Graph Translation

Graph translation refers to shifting the entire graph of a function to a different position on the coordinate plane. In the context of parabolas and quadratic functions, this typically involves horizontal and vertical translations.
For a function in vertex form, \( y = a(x-h)^2 + k \), the translations are controlled by \( h \) and \( k \):

• Changing \( h \) shifts the parabola left or right. If \( h \) increases, the graph shifts right; if \( h \) decreases, it shifts left.
• Changing \( k \) moves the parabola up or down. An increase in \( k \) moves it upwards, while a decrease shifts it downward.

Graph translations do not affect the shape or width of the parabola but merely reposition it on the graph.
This understanding helps in easily comparing different quadratic functions and determining their respective shifts.