Problem 19 - Algebra 2

Question

What is the effect on the graph of the equation \(y=x^{2}+2\) when the equation is changed to \(y=x^{2}-5 ?\)

Step-by-Step Solution

Answer

The graph shifts 7 units down.

Step 1: Identify Original Graph

The original equation is \(y = x^2 + 2\). This represents a parabola that opens upwards with vertex at the point \((0, 2)\). This is the standard form of a quadratic equation \(y = x^2\) shifted 2 units up along the y-axis.

Step 2: Identify New Graph

The new equation is \(y = x^2 - 5\). This is also a parabola that opens upwards, but its vertex is at the point \((0, -5)\). This equation represents a vertical shift of the original graph 7 units down.

Step 3: Determine the Effect of the Change

By comparing the two equations \(y = x^2 + 2\) and \(y = x^2 - 5\), we notice that the change from \(+2\) to \(-5\) in the constant term results in shifting the original graph downward by 7 units along the y-axis.

Key Concepts

Quadratic Functions

Quadratic functions are a fundamental concept in algebra and pre-calculus. They are defined by equations of the form \(y = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants. The graph of a quadratic function is called a parabola.
Parabolas are unique in that they have a symmetrical shape and can open upwards or downwards. If the coefficient \(a\) is positive, the parabola opens upwards, resembling a U-shape. If \(a\) is negative, it opens downwards. For any parabola, a key feature is its vertex, which represents the highest or lowest point on the graph, depending on the direction it opens.
Understanding the structure and transformation of quadratic functions is crucial as they not only apply to pure math problems but also to real-world phenomena such as projectile motion and optimizing areas and volumes.

Vertex Form

The vertex form of a quadratic function offers a convenient way to identify the vertex of the parabola. This form is expressed as \(y = a(x-h)^2 + k\), where \((h, k)\) is the vertex point of the parabola.
By using the vertex form, we can easily see how the graph translates on the coordinate plane. The value of \(h\) indicates a horizontal shift along the x-axis, while \(k\) shows a vertical shift along the y-axis.

• For example, an equation \(y = (x-3)^2 + 4\) corresponds to a parabola with its vertex at \((3, 4)\).
• Changes in \(a\) affect the width and direction of the parabola's opening.

Converting a quadratic equation to vertex form can greatly simplify understanding and predicting its graph's behavior and transformations.

Vertical Shift

Vertical shifts occur when we add or subtract a constant to or from the quadratic function. This transformation affects the graph by moving it up or down the y-axis without altering its shape.
A positive constant added moves the graph upwards, while a negative constant shifts it downwards. For instance, if you begin with the quadratic equation \(y = x^2 + 2\), adding 2 results in lifting the entire parabola two units upward, positioning the vertex at \((0, 2)\).
Alternatively, changing the equation to \(y = x^2 - 5\) indicates a vertical shift 7 units downward—this is because the graph moves from 2 units above the x-axis to 5 units below.

• Vertical shifts do not affect the x-coordinate of the vertex, only the y-coordinate.
• Such shifts maintain the 'U' shape of parabolas but reposition them vertically.

By recognizing vertical shifts, students can quickly sketch and interpret the movement of quadratic graphs.