If B^2-4ac 0 Which of the Following Conclusions Can Be Made About the Graph of F(X)=ax^2+bx+c

Quadratic functions are used in math and science to model various natural phenomena (such equally the height of falling objects).  However, there are some mutual questions about quadratic functions that it helps to know the answers to.

And so, what is at that place to know nigh quadratic functions? A quadratic function has 0, 1, or 2 10-intercepts, and it has the graph of a parabola that opens up or downward, not a straight line or periodic role. A quadratic function on a restricted domain can have an changed and as well both a maximum & minimum. No quadratic role has a negative exponent.

Of course, at that place are lots of difference scenarios for the graphs of quadratic functions (parabolas).  However, nosotros can allocate these into a few cases without likewise much trouble.

In this article, nosotros'll talk nigh quadratic functions in the context of 8 common questions about them.  Nosotros'll also wait at some graphs to illustrate some of the ideas.

Allow'south get started.

Questions Virtually Quadratic Functions

Quadratic functions raise lots of questions about the nature of their solutions, the appearance of their graphs, and their extreme (maximum and minimum values).

Let'south start off with a question concerning the solutions (or roots or zeros) of a quadratic role.

Can A Quadratic Function Have No X-Intercepts?

It is possible for a quadratic function to have no x-intercepts (too called zeros) if the corresponding quadratic equation has no real solutions.  Remember that this happens when the discriminant (the term b² – 4ac under the radical) in the quadratic formula is negative.

The quadratic formula. The discriminant is the term under the radical symbol in the numerator of the fraction.

In this instance, the 2 solutions to the quadratic equation are circuitous conjugates of the form a + bi and a – bi.  Since the solutions are complex with a nonzero imaginary part, they will non appear on a graph in the xy coordinate plane.

This means that the quadratic has no existent roots. Graphically, this means there are ii cases for a quadratic function f(ten) with no x-intercepts:

• The parabola is e'er above the x-axis.  In this case, nosotros must have a vertex above the x-axis and f''(x) > 0, pregnant f(10) is convex.
• The parabola is always beneath the x-centrality.  In this case, nosotros must have a vertex below the ten-axis and f''(10) < 0, meaning f(x) is concave.

You can run across some examples of quadratic functions with no x-intercepts below.

The graph of the parabola y = ten² + 5 has no ten-intercepts. It is ever above the x-axis (the line y = 0).

The graph of the parabola y = -xⁱⁱ – 3 has no 10-intercepts. It is ever below the x-axis

Notation that there are two other cases for the ten-intercepts of a quadratic part:

• A unmarried x-intercept – in this case, the vertex rests on the ten-centrality, and the x-coordinate of the vertex is a double (repeated) root of the quadratic.  This too means the discriminant is zero (that is, b² – 4ac = 0).  This quadratic has only ane real solution.
• Two singled-out x-intercepts – in this instance, the vertex does not rest on the x-axis, and at that place are ii roots of the quadratic: 1 to the left of the vertex, and one to the right of the vertex.  This as well ways the discriminant is positive (that is, b² – 4ac > 0).  This quadratic has two real solutions.

The graph of the parabola y = x^two has ane ten-intercept. Its vertex rests on the x-axis (the line y = 0).

The graph of the parabola f(x) = ii(x – three)(10 – 5) has two x-intercepts at x = 3 and x = 5. Its vertex is at x = four (halfway between the 2 zeros).

The graphic beneath summarizes the 3 possible cases for solutions of a quadratic equation, based on the sign of the discriminant.

The discriminant is the expression bⁱⁱ – 4ac under the radical in the quadratic formula. Its sign can tell the states the nature of the solutions of the corresponding quadratic equation.

Can A Quadratic Office Be A Direct Line?

A quadratic function cannot be a direct line.  By definition, a quadratic office has the class

• f(x) = ax² + bx + c

where a, b, and c are real numbers and a is not zero.

A quadratic has the class ax² + bx + c. The coefficients b and c may be zero, but a cannot be.

If nosotros have the graph of a straight line, it has the form bx + c.  This implies a = 0, which ways the graph is non a quadratic office.

The graph of the line y = 2x + 5. This is non a quadratic role, since a = 0 (that is, the x-squared term has a coefficient of zero).

Can A Quadratic Function Accept Three Zeros?

A quadratic function cannot have three zeros.  A quadratic office has exactly ii zeros (or roots), which may be existent or complex with nonzero imaginary parts.

A quadratic function has degree two, which means that the highest ability of 10 is ii (in the ax² term).  According to the Key Theorem of Algebra, a caste northward polynomial with complex coefficients has exactly n complex roots.

This ways that a quadratic function (with caste n = ii) has exactly ii roots.  We know the cases from earlier:

• Two distinct existent roots – this occurs when the discriminant bⁱⁱ – 4ac > 0.
• 1 repeated real root – also called a double root.  This occurs when the discriminant b² – 4ac = 0.
• Ii complex conjugate roots – this occurs when the discriminant b² – 4ac < 0.

This quadratic has two distinct real roots, meaning that the discriminant is positive.

This parabola has one repeated (double) real root, meaning that the discriminant is zero (and the vertex rests on the x-centrality).

This quadratic has no real solutions. This means that the discriminant is negative, and it has two complex conjugate roots.

Remember that each real null of a quadratic function represents a point where the graph intersects the x-axis (or the line y = 0).  These points are also known every bit x-intercepts.

Can A Quadratic Part Exist Periodic?

A quadratic role cannot exist periodic.  At almost, a quadratic function volition repeat a given y value twice (due to the degree of a quadratic being 2; see the Primal Theorem of Algebra before in this article).

A office similar f(10) = sin(10) is periodic because certain y values repeat indefinitely equally we take increasing x values.  For case, sin(x) has a value of 0 for whatsoever multiple of pi: sin(0), sin(π), sin(2π), sin(3π), … are all nada.

This is also true for negative multiples of pi: sin(-π), sin(-2π), sin(-3π), … are all cipher equally well.

You can see the graph of a periodic role sine function beneath.

A sine wave is periodic considering some y values repeat indefinitely as we continue to increment (or decrease) the value of ten.

Can A Quadratic Function Have A Maximum & A Minimum?

A quadratic function can have a maximum and a minimum if its domain is restricted (at to the lowest degree 1 closed endpoint].  For example, consider the quadratic office f(10) = ten^two + 3 on the domain [-ii, 4].

The minimum value of the part is at the vertex x = 0, which is in the interval [-2, 4].  The corresponding y value is 0² + 3 = iii.

The maximum value of the part is at the correct endpoint ten = 4 of the domain.  The corresponding y value is 4² + iii = 19.

The graph of the quadratic function f(x) = ten² + three on the domain [-2, four]. Its maximum is at the right endpoint (iv, 17) and its minimum is at the vertex (0, 3).

A quadratic function whose domain is the entire set of existent numbers can have merely a maximum or a minimum – not both.

Can A Quadratic Function Open Sideways?

A quadratic role defined by f(x) = ax² + bx + c cannot open sideways.  It will open upwards (if a > 0, meaning we accept a convex or concave upwardly function) or downwards (if a < 0, meaning we take a concave or concave down part).

A quadratic function of the grade f(10) = ax² + bx + c cannot open sideways. It tin only open up upwardly (for a > 0) or downwardly (for a < 0).

However, a quadratic divers by thousand(y) = ay² + by + c can open up sideways.  For example, the quadratic x = y² opens sideways (to the right).

A quadratic of the course x = y² tin open sideways.

Can A Quadratic Role Take A Negative Exponent?

A quadratic function cannot have negative exponents.  By definition, a quadratic has the form

• f(ten) = ax² + bx + c

This implies that for terms with nonzero coefficients, the exponents of the variable x tin can be 0, i, or 2.  There tin be no negative exponents in a quadratic function.

Can A Quadratic Function Have An Changed?

A quadratic part cannot have an changed if it is divers on the entire ready of real numbers.  This is due to the fact that a quadratic function is not 1-to-ane on a domain of the set of real numbers.

However, if we restrict its domain, a quadratic function can accept a well-defined changed.  For instance, take the quadratic office f(x) = 10² on the set of nonnegative real numbers (domain of x >= 0).

The inverse office would be f^-1(ten) = √x, divers on the prepare of nonnegative real numbers (domain of x >= 0).

The graph of the square root of 10, f^-1(10) = √x.

Conclusion

At present you know the answers to some commons questions most quadratic functions.  Yous also know how to reason through some of the ideas regarding the graphs of quadratic functions (parabolas).

You can learn more than about the roots and coefficients of a quadratic equation in my article here.

You might also want to check out my commodity on quadratic sequences.

You can also learn about quintic functions in my article here or my article on quartic functions in my article here.

You lot can also learn almost the connexion betwixt polynomials and functions in my article here.

I hope you found this article helpful.  If so, please share it with someone who can use the information.

Don't forget to subscribe to my YouTube aqueduct & go updates on new math videos!

~Jonathon

furnellprinnybod59.blogspot.com

Source: https://jdmeducational.com/questions-about-quadratic-functions-8-things-to-know/

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