math parabola
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Catégorie :Category: nCreator TI-Nspire
Auteur Author: joao.perrac
Type : Classeur 3.0.1
Page(s) : 1
Taille Size: 3.95 Ko KB
Mis en ligne Uploaded: 23/03/2025 - 13:19:10
Uploadeur Uploader: joao.perrac (Profil)
Téléchargements Downloads: 3
Visibilité Visibility: Archive publique
Shortlink : http://ti-pla.net/a4543657
Type : Classeur 3.0.1
Page(s) : 1
Taille Size: 3.95 Ko KB
Mis en ligne Uploaded: 23/03/2025 - 13:19:10
Uploadeur Uploader: joao.perrac (Profil)
Téléchargements Downloads: 3
Visibilité Visibility: Archive publique
Shortlink : http://ti-pla.net/a4543657
Description
Fichier Nspire généré sur TI-Planet.org.
Compatible OS 3.0 et ultérieurs.
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FACTOR FORM: y = a ( x - p ) ( x - q ) What each variable means: p and q are the x-intercepts, sometimes called the roots or zeros of the parabola. That means the parabola crosses the x-axis at x = p and at x = q. At those x-values, y = 0. a is the number that stretches or compresses the parabola, and also controls whether it opens up (a > 0) or down (a < 0). How to find p, q, and a from data: Identify the points where the parabola intersects the x-axis (if they are known). For example, if your data shows that y = 0 at x = 2 and x = 5, then p = 2 and q = 5. Once you have p and q, pick another data point (x1, y1) that is not on the x-axis (meaning y1 is not 0). Plug x1, y1 into y = a ( x - p ) ( x - q ). That means: y1 = a ( x1 - p ) ( x1 - q ). Solve for a. Now you have the complete equation. Example (illustration only) : Suppose you know the parabola crosses at x = 2 and x = 5 (so p = 2, q = 5). You also know a point (3, 6) lies on the parabola. Plug that in: 6 = a (3 - 2) (3 - 5).(3 - 2) = 1 and (3 - 5) = -2, so the product is -2.Therefore, 6 = a * -2, which gives a = -3. The parabola is then y = -3 ( x - 2 ) ( x - 5 ). This form is especially handy when your data clearly shows two x-intercepts (p and q). If you only have approximate intercepts or no real intercepts, factor form is trickier to set up directly. VERTEX FORM: y = a ( x - h )^2 + k What each variable means: (h, k) is the vertex of the parabola. That is, h is the x-value of the vertex, and k is the y-value of the vertex. a is again the vertical stretch factor. A positive a means the parabola opens upward, a negative a means it opens downward. Larger absolute values of a make the parabola narrower, smaller absolute values of a make it wider. In this form: If the parabola opens upward (a > 0), the vertex is a minimum point at (h, k). If the parabola opens downward (a < 0), the vertex is a maximum point at (h, k). How to find a, h, and k from data: It depends on what you know: A) If the vertex is known. Suppose you know the vertex is at (h, k). That often comes from seeing where the parabolas turning point is in your data. Then pick another data point (x1, y1) that lies on the parabola but is not the vertex. Plug x1, y1 into y = a ( x - h )^2 + k. In other words: y1 = a ( x1 - h )^2 + k. Solve for a. Once you have a, h, and k, your equation is complete. Example (illustration only) : Suppose from the data you know the vertex is at (4, 2). That means h = 4, k = 2. You also know that the point (6, 8) is on the parabola. Then you do 8 = a (6 - 4)^2 + 2. Here, (6 - 4) = 2, and 2 squared = 4, so the right side is a * 4 + 2. So 8 = 4a + 2, meaning 4a = 6, giving a = 1.5. That means y = 1.5 ( x - 4 )^2 + 2. B) If you do not directly know the vertex. You can still find h and k by using multiple points in a system of equations. Generally, you need at least three points (x1, y1), (x2, y2), (x3, y3) to solve for a, h, and k. Youd set up: y1 = a ( x1 - h )^2 + k y2 = a ( x2 - h )^2 + k y3 = a ( x3 - h )^2 + k Solve the system to find a, h, and k. (This is more algebra and is typically done if the vertex is not obvious.) Important Note: In many real-data problems, you might do a best fit approach with a calculator or other methods if your data does not lie exactly on a perfect parabola. But if it does, or if youre told its definitely parabolic, these steps show how to get a formula from the data. EXTRA TIPS Plot your data first, if possible. It helps you see if the parabola definitely crosses the x-axis (making factor form easy), or if the vertex is visible (making vertex form easier). Check extra points if you have them. After finding your equation, plug in other data points to see if they match well. If they do not, you might have to refine your approach or consider that the data is only approximately parabolic. Understanding a : If a is positive, the parabola opens upward (U-shape). If a is negative, it opens downward (an upside-down U). What about standard form (ax^2 + bx + c)? That is another approach: you can also solve for a, b, and c from your data. Factor form and vertex form, however, can be faster if you directly know intercepts or the vertex location. Why might I need vertex form y = a ( x - h )^2 + k? If a real-world problem says the parabolas highest point is at x = 10, y = 50, that is basically giving you (h, k) = (10, 50). Then you find a using another known data point. Why might I need factor form y = a ( x - p ) ( x - q )? If you know the parabola crosses the x-axis at x = p and x = q, then that is a quick way to build the equation. You just need a third point to solve for a. Made with nCreator - tiplanet.org
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Compatible OS 3.0 et ultérieurs.
<<
FACTOR FORM: y = a ( x - p ) ( x - q ) What each variable means: p and q are the x-intercepts, sometimes called the roots or zeros of the parabola. That means the parabola crosses the x-axis at x = p and at x = q. At those x-values, y = 0. a is the number that stretches or compresses the parabola, and also controls whether it opens up (a > 0) or down (a < 0). How to find p, q, and a from data: Identify the points where the parabola intersects the x-axis (if they are known). For example, if your data shows that y = 0 at x = 2 and x = 5, then p = 2 and q = 5. Once you have p and q, pick another data point (x1, y1) that is not on the x-axis (meaning y1 is not 0). Plug x1, y1 into y = a ( x - p ) ( x - q ). That means: y1 = a ( x1 - p ) ( x1 - q ). Solve for a. Now you have the complete equation. Example (illustration only) : Suppose you know the parabola crosses at x = 2 and x = 5 (so p = 2, q = 5). You also know a point (3, 6) lies on the parabola. Plug that in: 6 = a (3 - 2) (3 - 5).(3 - 2) = 1 and (3 - 5) = -2, so the product is -2.Therefore, 6 = a * -2, which gives a = -3. The parabola is then y = -3 ( x - 2 ) ( x - 5 ). This form is especially handy when your data clearly shows two x-intercepts (p and q). If you only have approximate intercepts or no real intercepts, factor form is trickier to set up directly. VERTEX FORM: y = a ( x - h )^2 + k What each variable means: (h, k) is the vertex of the parabola. That is, h is the x-value of the vertex, and k is the y-value of the vertex. a is again the vertical stretch factor. A positive a means the parabola opens upward, a negative a means it opens downward. Larger absolute values of a make the parabola narrower, smaller absolute values of a make it wider. In this form: If the parabola opens upward (a > 0), the vertex is a minimum point at (h, k). If the parabola opens downward (a < 0), the vertex is a maximum point at (h, k). How to find a, h, and k from data: It depends on what you know: A) If the vertex is known. Suppose you know the vertex is at (h, k). That often comes from seeing where the parabolas turning point is in your data. Then pick another data point (x1, y1) that lies on the parabola but is not the vertex. Plug x1, y1 into y = a ( x - h )^2 + k. In other words: y1 = a ( x1 - h )^2 + k. Solve for a. Once you have a, h, and k, your equation is complete. Example (illustration only) : Suppose from the data you know the vertex is at (4, 2). That means h = 4, k = 2. You also know that the point (6, 8) is on the parabola. Then you do 8 = a (6 - 4)^2 + 2. Here, (6 - 4) = 2, and 2 squared = 4, so the right side is a * 4 + 2. So 8 = 4a + 2, meaning 4a = 6, giving a = 1.5. That means y = 1.5 ( x - 4 )^2 + 2. B) If you do not directly know the vertex. You can still find h and k by using multiple points in a system of equations. Generally, you need at least three points (x1, y1), (x2, y2), (x3, y3) to solve for a, h, and k. Youd set up: y1 = a ( x1 - h )^2 + k y2 = a ( x2 - h )^2 + k y3 = a ( x3 - h )^2 + k Solve the system to find a, h, and k. (This is more algebra and is typically done if the vertex is not obvious.) Important Note: In many real-data problems, you might do a best fit approach with a calculator or other methods if your data does not lie exactly on a perfect parabola. But if it does, or if youre told its definitely parabolic, these steps show how to get a formula from the data. EXTRA TIPS Plot your data first, if possible. It helps you see if the parabola definitely crosses the x-axis (making factor form easy), or if the vertex is visible (making vertex form easier). Check extra points if you have them. After finding your equation, plug in other data points to see if they match well. If they do not, you might have to refine your approach or consider that the data is only approximately parabolic. Understanding a : If a is positive, the parabola opens upward (U-shape). If a is negative, it opens downward (an upside-down U). What about standard form (ax^2 + bx + c)? That is another approach: you can also solve for a, b, and c from your data. Factor form and vertex form, however, can be faster if you directly know intercepts or the vertex location. Why might I need vertex form y = a ( x - h )^2 + k? If a real-world problem says the parabolas highest point is at x = 10, y = 50, that is basically giving you (h, k) = (10, 50). Then you find a using another known data point. Why might I need factor form y = a ( x - p ) ( x - q )? If you know the parabola crosses the x-axis at x = p and x = q, then that is a quick way to build the equation. You just need a third point to solve for a. Made with nCreator - tiplanet.org
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