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Chapter 11: Problem 5

Which equations have a graph that is a vertical parabola? A horizontal parabola? A. \(y=-x^{2}+20 x+80\) B. \(x=2 y^{2}+6 y+5\) C. \(x+1=(y+2)^{2}\) D. \(f(x)=(x-4)^{2}\)

Short Answer

Expert verified
Vertical parabolas: A and D. Horizontal parabolas: B and C.

Step by step solution

01

- Identify the form of the equation

The general form of a vertical parabola is given by \(y = ax^2 + bx + c\) or \(y = a(x - h)^2 + k\). For a horizontal parabola, the general form is \(x = ay^2 + by + c\) or \(x = a(y - k)^2 + h\).
02

- Analyze each equation

Determine the form of each given equation: A. \(y = -x^{2} + 20x + 80\) is in the form of \(y = ax^2 + bx + c\), so it is a vertical parabola. B. \(x = 2y^{2} + 6y + 5\) is in the form of \(x = ay^2 + by + c\), so it is a horizontal parabola. C. \(x + 1 = (y + 2)^2\) can be rewritten as \(x = (y + 2)^2 - 1\), which is in the form of \(x = a(y - k)^2 + h\), indicating a horizontal parabola. D. \(f(x) = (x - 4)^2\) can be rewritten as \(y = (x - 4)^2\), which is in the form of \(y = a(x - h)^2 + k\), indicating a vertical parabola.
03

- Categorize the equations

From the analysis: Vertical parabolas: A and D. Horizontal parabolas: B and C.

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Key Concepts

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

Vertical Parabolas
In algebra, parabolas that open upwards or downwards are called vertical parabolas. These parabolas have equations in the form of \(y = ax^2 + bx + c\) or \(y = a(x - h)^2 + k\). The terms and structure of the equation help to determine its orientation and shape.
  • Quadratic Form (\(y = ax^2 + bx + c\)): When a parabola鈥檚 equation looks like this, it opens up if \(a > 0\) and down if \(a < 0\).
  • Vertex Form (\(y = a(x - h)^2 + k\)): This shows the vertex, or the highest or lowest point of the parabola at \((h, k)\). It also reveals the direction it opens, based on the sign of \(a\).

For example, let鈥檚 look at the equations given in the exercise:
  • Equation A: \(y = -x^{2} + 20x + 80\) 鈥� It's in the form \(y = ax^2 + bx + c\), with \(a = -1\). It鈥檚 a vertical parabola that opens downwards.
  • Equation D: \(f(x) = (x - 4)^2\) 鈥� It can be rewritten as \(y = (x - 4)^2\), which is in the vertex form \(y = a(x - h)^2 + k\) with \(a = 1\). It鈥檚 a vertical parabola that opens upwards.
Horizontal Parabolas
Parabolas that open to the left or right are called horizontal parabolas. Their equations typically have the form \(x = ay^2 + by + c\) or \(x = a(y - k)^2 + h\). The orientation and specific shape can be determined by understanding the equation's structure.
  • Quadratic Form (\(x = ay^2 + by + c\)): This indicates a parabola that opens to the right if \(a > 0\) and to the left if \(a < 0\).
  • Vertex Form (\(x = a(y - k)^2 + h\)): Displays the vertex at \((h, k)\) and shows the direction it opens, depending on the sign of \(a\).

Looking at the given equations:
  • Equation B: \(x = 2y^{2} + 6y + 5\) 鈥� This is in the form \(x = ay^2 + by + c\) with \(a = 2\). It's a horizontal parabola that opens to the right.
  • Equation C: \(x + 1 = (y + 2)^2\) 鈥� Rewrite to \(x = (y + 2)^2 - 1\). It fits the form \(x = a(y - k)^2 + h\), indicating it鈥檚 a horizontal parabola that opens to the right.
Equation Forms
Understanding the form of parabolic equations is crucial. They come in two primary forms: quadratic form and vertex form. Each helps to identify key characteristics of a parabola鈥檚 graph.
  • Quadratic Form: Equations like \(y = ax^2 + bx + c\) or \(x = ay^2 + by + c\) show general quadratic relationships. You can find the vertex and the direction of the opening by using the coefficient \(a\).
  • Vertex Form: Equations such as \(y = a(x - h)^2 + k\) or \(x = a(y - k)^2 + h\) reveal the vertex directly at \((h, k)\). The constant \(a\) still shows the direction of the opening (upward, downward, left, or right).

When analyzing a parabolic equation, always try to rewrite it according to these standard forms to easily understand its properties. Like in our examples, rewriting equations C and D helped to categorize their directions correctly. This approach simplifies understanding and graphing parabolas.

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Most popular questions from this chapter

In the 1939 classic movie The Wizard of Oz, Ray Bolger's character, the Scarecrow, wants a brain. When the Wizard grants him his "Th.D." (Doctor of Thinkology), the Scarecrow replies with the following statement. Scarecrow: The sum of the square roots of any two sides of an isosceles triangle is equal to the square root of the remaining side. His statement sounds like the formula for the Pythagorean theorem. To see why it is incorrect, In the Scarecrow's statement, he refers to square roots. In applying the formula for the Pythagorean theorem, do we find square roots of the sides? If not, what do we find?

Solve for \(x .\) Assume that a and \(b\) represent positive real numbers. \(9 x^{2}-25 a=0\)

Find the discriminant. Use it to determine whether the solutions for each equation are A. two rational numbers B. one rational number C. two irrational numbers D. two nonreal complex numbers. Tell whether the equation can be solved using the zero-factor property, or if the quadratic formula should be used instead. Do not actually solve. $$ x^{2}+4 x+2=0 $$

Solve using the square root property. Simplify all radicals. $$ (3-2 x)^{2}=70 $$

Solve each equation. (All solutions are nonreal complex numbers.) $$ x^{2}=-21 $$
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