Mr. Haas
Graphing Quadratic Functions: Introduction. Graphing Quadratic Functions (page 1 of 4) Sections: Introduction, The meaning of the leading coefficient / The vertex, Examples The general technique for graphing quadratics is the same as for graphing linear equations. However, since quadratics graph as curvy lines (called "parabolas"), rather than the straight lines generated by linear equations, there are some additional considerations. The most basic quadratic is y = x2. When you graphed straight lines, you only needed two points to graph your line, though you generally plotted three or more points just to be on the safe side.
He got the graph wrong. That last point has a rather large y-value, so you decide that you won't bother drawing your graph large enough to plot it. Even if you'd forgotten that quadratics graph as curvy parabolas, these points will remind you of this fact. Unlike the careless student, you just got the graph right. This is not correct. Top | 1 | 2 | 3 | 4 | Return to Index Next >> Graphing Quadratic Equations. A Quadratic Equation in Standard Form (a, b, and c can have any value, except that a can't be 0.) Here is an example: Graphing You can graph a Quadratic Equation using the Function Grapher, but to really understand what is going on, you can make the graph yourself. Read On! The Simplest Quadratic The simplest Quadratic Equation is: f(x) = x2 And its graph is simple too: This is the curve f(x) = x2 It is a parabola. Now let us see what happens when we introduce the "a" value: f(x) = ax2 Larger values of a squash the curve inwards Smaller values of a expand it outwards And negative values of a flip it upside down The "General" Quadratic Before graphing we rearrange the equation, from this: f(x) = ax2 + bx + c To this: f(x) = a(x-h)2 + k Where: h = -b/2a k = f( h ) In other words, calculate h (=-b/2a), then find k by calculating the whole equation for x=h First of all ...
So ... Lets see an example of how to do this: Example: Plot f(x) = 2x2 - 12x + 16 First, let's note down: a = 2, b = -12, and c = 16. How to Graph a Quadratic Equation: 10 Steps. Edit Article Community Q&A When graphed, quadratic equations of the form ax2 + bx + c or a(x - h)2 + k give a smooth U-shaped or a reverse U-shaped curve called a parabola. Graphing a quadratic equation is a matter of finding its vertex, direction, and, often, its x and y intercepts. In the cases of relatively simple quadratic equations, it may also be enough to plug in a range of x values and plot a curve based on the resulting points.
See Step 1 below to get started. Ad Steps <img alt="Image titled Graph a Quadratic Equation Step 1" src=" width="728" height="546" class="whcdn" onload="WH.performance.clearMarks('image1_rendered'); WH.performance.mark('image1_rendered');">1Determine which form of quadratic equation you have. Can you answer these readers' questions? How long do I have to roll over a balance into another plan Tips Article Info Categories: Algebra. 3 Ways to Solve Quadratic Equations. How to solve a quadratic equation by factoring (example) | Solving quadratics by factoring. Completing the square. The quadratic formula - A complete course in algebra. Table of Contents | Home Back to Section 1 Completing the square The quadratic formula The discriminant Proof of the quadratic formula IN LESSON 18 we saw a technique called completing the square.
Completing the square If we try to solve this quadratic equation by factoring, This technique is valid only when the coefficient of x2 is 1. 1) Transpose the constant term to the right x2 + 6x = −2. x2 + 6x + 9 = −2 + 9. The left-hand side is now the perfect square of (x + 3). (x + 3)2 = 7. 3 is half of the coefficient 6. That equation has the form That is, the solutions to x2 + 6x + 2 = 0 are the conjugate pair, For a method of checking these roots, see the theorem of the sum and product of the roots: Lesson 10 of Topics in Precalculus, In Lesson 18 there are examples and problems in which the coefficient of x is odd. Problem 6. To see the answer, pass your cursor from left to rightover the colored area. Problem 7. The quadratic formula Here is a formula for finding the roots of any quadratic. Theorem. Then. Factoring Quadratics: The Simple Case. Factoring Quadratics: The Simple Case (page 1 of 4) Sections: The simple case, The hard case, The weird case A "quadratic" is a polynomial that looks like "ax2 + bx + c", where "a", "b", and "c" are just numbers.
For the easy case of factoring, you will find two numbers that will not only multiply to equal the constant term "c", but also add up to equal "b", the coefficient on the x-term. For instance: Factor x2 + 5x + 6. I need to find factors of 6 that add up to 5. Since 6 can be written as the product of 2 and 3, and since 2 + 3 = 5, then I'll use 2 and 3. (x )(x ) Then I'll write in the two numbers that I found above: (x + 2)(x + 3) This is the answer: x2 + 5x + 6 = (x + 2)(x + 3) This is how all of the "easy" quadratics will work: you will find factors of the constant term that add up to the middle term, and use these factors to fill in your parentheses.
Your text or teacher may refer to factoring "by grouping", which is covered in the lesson on simple factoring. Factor x2 + 7x + 6. Factoring Quadratics: The Hard Case. Factoring Quadratics: The Hard Case: The Modified "a-b-c" Method, or "Box" (page 2 of 4) Sections: The simple case, The hard case, The weird case To factor a "hard" quadratic, we have to handle all three coefficients, not just the two we handled in the "easy" case, because the leading coefficient (the number on the x2 term) is not 1. The first step in factoring will be to multiply "a" and "c"; then we'll need to find factors of the product "ac" that add up to "b".
Factor 2x2 + x– 6. Looking at this quadratic, I have a = 2, b = 1, and c = –6, so ac = (2)(–6) = –12. Then I will take my factors –3 and 4 and put them, complete with their signs and variables, in the diagonal corners, like this: (It doesn't matter which way you do the diagonal entries; the answer will work out the same either way!) Then I'll factor the rows and columns like this: (Note: The signs for the bottom-row entry and the right-column entry come from the closest term that you are factoring from. Factor 4x2 – 19x + 12. Integrated Math I.