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Magazine Chapter 10: Problem 21
Sketch the graph of the given equation. $$ (y-1)^{2}=16 $$
Short Answer
Expert verified
The graph is a horizontal line opening at \((0, 1)\) with endpoints \((-4, 1)\) and \((4, 1)\).
Step by step solution
01
Identify the Type of Equation
The given equation is \((y-1)^2 = 16\). This equation can be rewritten as \((y-1)^2 = 4^2\). It represents a horizontal parabola because the equation is in the form \((y-k)^2 = 4p(x - h)\), which is the standard form for a parabola that opens horizontally.
02
Determine the Vertex of the Parabola
In the equation \((y-1)^2 = 16\), it indicates that there is no \(x\) term, which implies the formula is centered horizontally at \(x = 0\). The \(y\) is translated up by 1 unit, so the vertex of the parabola is at \((0, 1)\).
03
Determine the Direction the Parabola Opens
Since the equation is written in the form \((y-1)^2 = 4^2\), the square of a real number implies that the parabola opens horizontally. Specifically, because there is no negative sign in front of the equation, the parabola opens to both sides, resembling a horizontal line.
04
Plot Key Points
To sketch the graph, first plot the vertex at \((0, 1)\). Since the equation states \((y-1)^2 = 4^2\), the points to plot along the \(x\)-axis are 4 units to the right and left of the vertex, resulting in points \((4, 1)\) and \((-4, 1)\).
05
Sketch the Parabola
Draw a smooth curve passing through the points \((4, 1)\), \((0, 1)\), and \((-4, 1)\). Ensure the parabola is symmetric about the horizontal line \(y = 1\), resembling a horizontal line extending infinitely to both sides.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Parabola
A parabola is a U-shaped graph that can open either vertically or horizontally, based on its equation. It is the graphical representation of a quadratic equation. The standard forms of a parabola are
- For vertical parabolas: \( (x-h)^2 = 4p(y-k) \)
- For horizontal parabolas: \( (y-k)^2 = 4p(x-h) \)
Vertex
The vertex of a parabola is the point where it changes direction.
- For a vertical parabola, the vertex is where the graph turns at its highest or lowest point. The parabola will open upwards if \( 4p > 0 \) and downwards if \( 4p < 0 \).
- In contrast, the vertex for a horizontal parabola represents the leftmost or rightmost point. The equation of our exercise, \( (y-1)^2 = 16 \), indicates a vertex at \( (0, 1) \).
Symmetry
Symmetry in a parabola signifies that one half of the parabola is a mirror image of the other. This is an important feature because it helps in sketching and predicting the parabola鈥檚 path.
- For a vertical parabola, the symmetry axis is a vertical line through the vertex, typically \( x = h \).
- For a horizontal parabola, as in the given equation, the symmetry axis is a horizontal line, \( y = k \).
Horizontal Parabola
A horizontal parabola has distinct traits because it opens outwards sideways rather than up and down. Its general form, as identified previously, is \( (y-k)^2 = 4p(x-h) \). This shows two outcomes:
- The direction in which the parabola opens, left or right, depends on the sign of \( p \). If \( p > 0 \), it opens rightwards.
- If \( p < 0 \), it opens towards the left.
