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Chapter 2: Problem 50

Write an equation in standard form of the parabola that has the same shape as the graph of \(f(x)=2 x^{2},\) but with the given point as the vertex. $$(7,4)$$

Short Answer

Expert verified
The equation in standard form of the parabola that has the same shape as the graph of \(f(x) = 2 x^2\), but with the vertex at (7, 4) is \(y = 2(x - 7)^2 + 4\).

Step by step solution

01

Identify Vertex Coordinates

The given vertex is (7, 4). This means h = 7 and k = 4, where (h, k) are the coordinates of the vertex.
02

Recall the Standard Form of a Parabola

The standard form of a parabola is \(y = a(x - h)^2 + k\), where (h, k) are the vertex's coordinates and a is the coefficient that determines the parabola's orientation.
03

Substitute Vertex and coefficient values into the Standard Form

Substitute h = 7, k = 4 and a = 2 into the standard form equation to get \(y = 2(x - 7)^2 + 4\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Vertex Form of a Parabola
The vertex form of a parabola is a powerful way of expressing quadratic equations that emphasizes the parabola's vertex. In this form, the equation is written as \( y = a(x-h)^2 + k \). Here,
  • \((h, k)\): Represents the vertex coordinates
  • \(a\): Determines the "width" and "direction" of the parabola (upwards if \(a > 0\), downwards if \(a < 0\))
This form is particularly useful because it allows you to quickly identify the parabola's vertex, making it easier to graph. If you know the vertex, you can immediately see where the parabola's peak or trough is located. This can simplify solving problems that involve maximum or minimum values.
Standard Form of a Parabola
The standard form of a parabola helps understand the shape and direction of a parabola's graph. The general equation for a parabola in standard form is \( ax^2 + bx + c = y \). In this form,
  • \(a\): Still affects the "width" and "direction" of the parabola
  • \(b\) and \(c\): Influence the position of the graph along the x-axis and y-axis
Converting from vertex form to standard form, you expand the squared term and distribute \(a\) through the equation. For instance, starting with \(y = a(x-h)^2 + k\), you would expand \((x-h)^2\) and multiply by \(a\), and then combine constants and like terms to obtain \(ax^2 + bx + c\). This form is often used for analyzing the symmetry of the graph.
Graph Transformation
Graph transformations modify the original graph of a function in definite ways, enhancing its visualization. They can involve shifts, stretches, or reflections.
  • Horizontal Shift: Adjusts the graph left or right. This is seen directly in the \(x-h\) part of the vertex form.
  • Vertical Shift: Moves the graph up or down. This is captured by \(k\) in the vertex form.
  • Vertical Stretch/Compression: Determines the "tightness" or "width" of the parabola. A larger \(|a|\) value makes the parabola narrower, while smaller \(|a|\) values make it wider.
Reflecting over the x-axis changes the parabola's orientation (up vs. down), achieved by altering the sign of \(a\). By understanding these transformations, you can predict and graph multiple forms of parabolas quickly.

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