![]() Usually, to make the equation as simple as possible, we try to search for the greatest common factor. Using your knowledge of how to factor both lone numbers and variables with coefficients, you can simplify simple algebraic equations by finding factors that the numbers and variables in an algebraic equation have in common. This article has been viewed 661,680 times.Īpply the distributive property of multiplication to factor algebraic equations. This article has been fact-checked, ensuring the accuracy of any cited facts and confirming the authority of its sources. There are 7 references cited in this article, which can be found at the bottom of the page. Additionally, David has worked as an instructor for online videos for textbook companies such as Larson Texts, Big Ideas Learning, and Big Ideas Math. ![]() After attaining a perfect 800 math score and a 690 English score on the SAT, David was awarded the Dickinson Scholarship from the University of Miami, where he graduated with a Bachelor’s degree in Business Administration. With over 10 years of teaching experience, David works with students of all ages and grades in various subjects, as well as college admissions counseling and test preparation for the SAT, ACT, ISEE, and more. David Jia is an Academic Tutor and the Founder of LA Math Tutoring, a private tutoring company based in Los Angeles, California. This article was co-authored by David Jia. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution License. Use the information below to generate a citation. Then you must include on every digital page view the following attribution: If you are redistributing all or part of this book in a digital format, Then you must include on every physical page the following attribution: If you are redistributing all or part of this book in a print format, Want to cite, share, or modify this book? This book uses the This book may not be used in the training of large language models or otherwise be ingested into large language models or generative AI offerings without OpenStax's permission. We can see this by expanding out the general form and setting it equal to the standard form. The standard form and the general form are equivalent methods of describing the same function. But if | a | 1, | a | > 1, so the graph becomes narrower. If h > 0, h > 0, the graph shifts toward the right and if h 1, | a | > 1, the point associated with a particular x - x - value shifts farther from the x-axis, so the graph appears to become narrower, and there is a vertical stretch. If k > 0, k > 0, the graph shifts upward, whereas if k 0, k > 0, so the graph is shifted 4 units upward. The x - x - intercepts, those points where the parabola crosses the x - x - axis, occur at ( −3, 0 ) ( −3, 0 ) and ( −1, 0 ). For a parabola that opens upward, the vertex occurs at the lowest point on the graph, in this instance, ( −2, −1 ). The vertex always occurs along the axis of symmetry. This also makes sense because we can see from the graph that the vertical line x = −2 x = −2 divides the graph in half. The axis of symmetry is x = − 4 2 ( 1 ) = −2. If a 0, a > 0, the parabola opens upward. If a > 0, a > 0, the parabola opens upward. Where a, b, a, b, and c c are real numbers and a ≠ 0. These features are illustrated in Figure 2.į ( x ) = a x 2 + b x + c f ( x ) = a x 2 + b x + c The graph is also symmetric with a vertical line drawn through the vertex, called the axis of symmetry. In either case, the vertex is a turning point on the graph. If the parabola opens down, the vertex represents the highest point on the graph, or the maximum value. If the parabola opens up, the vertex represents the lowest point on the graph, or the minimum value of the quadratic function. One important feature of the graph is that it has an extreme point, called the vertex. The graph of a quadratic function is a U-shaped curve called a parabola. Working with quadratic functions can be less complex than working with higher degree functions, so they provide a good opportunity for a detailed study of function behavior. In this section, we will investigate quadratic functions, which frequently model problems involving area and projectile motion. The cross-section of the antenna is in the shape of a parabola, which can be described by a quadratic function. (credit: Matthew Colvin de Valle, Flickr)Ĭurved antennas, such as the ones shown in Figure 1, are commonly used to focus microwaves and radio waves to transmit television and telephone signals, as well as satellite and spacecraft communication.
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