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Chapter 11: Problem 62

Is it possible for the graph of a quadratic function to have only one \(x\) -intercept if the vertex is off the \(x\) -axis? Why or why not?

Short Answer

Expert verified
No, a quadratic function's graph cannot have only one x-intercept if the vertex is off the x-axis.

Step by step solution

01

- Understand the problem

Identify the key elements of the problem: a quadratic function graphed on the Cartesian plane and its potential number of x-intercepts.
02

- Recall the properties of a quadratic function

A quadratic function is generally in the form of \( f(x) = ax^2 + bx + c \). The graph of such a function is a parabola, which can intersect the x-axis at 0, 1, or 2 points.
03

- Define the vertex of a quadratic function

The vertex of a parabola given by \( f(x) = ax^2 + bx + c \) is at the point \( (h, k) \), where \( h = -\frac{b}{2a} \) and \( k = f(h) \).
04

- Consider the vertex position relative to the x-axis

The problem states that the vertex is off the x-axis, meaning \( k eq 0 \).
05

- Analyze the number of x-intercepts

For the parabola to have exactly one x-intercept, the vertex must lie on the x-axis (\( k = 0 \)). If the vertex is not on the x-axis (\( k eq 0 \)), the parabola can either intersect the x-axis at two points or not intersect at all, depending on whether \( ax^2 \) opens upwards or downwards.
06

- Conclusion

If the vertex is off the x-axis (\( k eq 0 \)), it is impossible for the parabola to have exactly one x-intercept.

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Key Concepts

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

quadratic functions
A quadratic function is a specific type of polynomial function with the general form \( f(x) = ax^2 + bx + c \), where \( a, b, \) and \( c \) are constants and \( a eq 0 \). The graph of a quadratic function is a curve called a parabola. The parabola can open upwards (if \( a > 0\)) or downwards (if \( a < 0\)).

Key characteristics of quadratic functions include:
  • **Vertex:** This is the highest or lowest point on the graph, depending on the parabola's direction.
  • **Axis of symmetry:** A vertical line through the vertex, given by \( x = h \), divides the parabola into two symmetrical halves.
  • **X-intercepts:** Points where the graph intersects the x-axis.
  • **Y-intercept:** The point where the graph intersects the y-axis (where \( x = 0 \)).
Understanding these elements will help you analyze the behavior and properties of quadratic functions.
parabola vertex
The vertex of a parabola is a crucial point where the graph changes direction. For a quadratic function \( f(x) = ax^2 + bx + c \), the vertex is located at the point \( (h, k) \).

Here's how you can find the vertex:
  • First, calculate \( h \) using the formula \( h = -\frac{b}{2a} \).
  • Then, find \( k \) by substituting \( h \) back into the function: \( k = f(h) \).
The coordinates \( (h, k) \) give you the exact location of the vertex, which helps in determining the shape and position of the parabola.

If the vertex is located on the x-axis, it means \( k = 0 \). This condition is essential when considering the number of x-intercepts a parabola can have.
x-intercepts
X-intercepts are the points at which the graph of the quadratic function crosses the x-axis. These are also known as the 'roots' or 'zeroes' of the function. To find the x-intercepts of a quadratic function \( f(x) = ax^2 + bx + c \), solve the equation \( ax^2 + bx + c = 0 \).

There are three possible scenarios:
  • The parabola has **two distinct x-intercepts** if the discriminant \( b^2 - 4ac \) is greater than 0.
  • The parabola has **one x-intercept** (it touches the x-axis at one point) if the discriminant is equal to 0. This can only happen when the vertex is on the x-axis.
  • The parabola has **no x-intercepts** if the discriminant is less than 0.
In the given problem, since the vertex is off the x-axis \( k eq 0 \), it is impossible for the parabola to have only one x-intercept. This is because a parabola can have exactly one x-intercept if and only if its vertex lies directly on the x-axis.

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Most popular questions from this chapter

Let \(g(x)=x^{2}+14 x+49 .\) Find \(x\) such that \(g(x)=49\).

Review finding intercepts and completing the square (Sections \(3.3,6.2,6.3, \text { and } 11.1)\) Find the \(x\) -intercepts. $$ y=2 x^{2}-x-3 $$

Solve by completing the square. Show your work. $$ t^{2}+8 t-3=0 $$

Without graphing, find the vertex, the axis of symmetry, and the maximum value or the minimum value. $$ f(x)=\sqrt{2}(x+4.58)^{2}+65 \pi $$

Find the \(x\) -intercepts of the function given by \(f(x)=4 x^{2}+2 x-3\).
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