![]() Now set two pairs of equations equal to one another, eliminating C: equations 1 and 2, 2 and 3: If we solve all three equations for C, we get: We first use two sets of two equations to eliminate one of the variables ( C is usually easiest in these problems), then use those two equations to solve for the other two, and finally use the results to find the third variable. These three equations can be solved simultaneously by methods you already know. Notice that none of these appears to be the vertex of the parabola, but that won't matter. It's probably best at this point to work an example or two, so let's find the equation of the parabola above, which passes through (-4, 8), (1, -4) and (3, -2). Now that's three equations and three unknowns ( A, B, C), so we should have enough information to determine what they are. If each of out three points satisfies this equation for a certain set of A, B and C (the things we're really looking for), we have three equations: Let's say that a parabola passes through three known points, $(x_1, y_1), (x_2, y_2),$ and $(x_3, y_3).$ Now we know that the general form of the equation we're looking for is We'd like to be able to find the equation of that parabola. It's a very specific kind of curve, the graph of a quadratic function. That may be a bit hard to come to terms with at first, but remember that a parabola isn't just any curve. Nicole drops the ball from 4 feet above the ground, so the point representing the \(y\)-intercept is \((0,4)\).Through any three points, only one unique parabola can be drawn. Since the value of \(c\) in the expanded equation is 4, the \(y\)-intercept is 4. Since both binomials are the same, there is one solution, which is 2. Next, set each binomial equal to 0 and solve for \(x\). Start by writing the quadratic equation as two binomials. Since the length of the flag is 4 feet longer than its width, the length of the flag is 12 feet. Therefore, the width of the flag is 8 feet. Since this problem is about the area of a flag, the negative solution, -12, does not apply to this scenario. Plot the vertex and the \(x\)-intercepts onto the coordinate plane and join the points with a smooth curve. The vertex of this function is \((-2,-100)\). Substitute -2 into the quadratic equation for \(w\) and simplify: Next, find \(k\), which is the vertex’s \(y\)-coordinate. Divide the sum of the \(x\)-intercepts by 2: Start by finding \(h\), which is the vertex’s \(x\)-coordinate. Now that we know the \(x\)-intercepts, find the coordinates for the vertex, \((h,k)\). The graph of the function passes through the \(x\)-axis at -12 and 8. These numbers are 12 and -8.įrom here, equate each binomial to 0 and solve for \(w\). Factor the equation by finding two numbers that result in a sum of 4 and a product of -96. Then, identify the coordinates for the \(x\)-intercepts. Write the quadratic equation in standard form. Next, simplify the equation by distributing \(w\). Since the length of the flag is 4 feet longer than its width, use \(w 4\) to represent the length. Since the width is not known, use w to represent width. Substitute the values from the word problem into this formula. Start by recalling the formula for the area of a rectangle, which is length times width. The vertex is the point on the parabola where the graph intersects its axis of symmetry. However, all parabolas share the same U-shape.Ī parabola is symmetric over an invisible line called the axis of symmetry. The parabola can open upward or downward and can vary in width. The graph of a quadratic function is a two-dimensional curve called a parabola. First, a quadratic function is a polynomial function, and its highest degree term is of the second degree. We’ll also talk about how to graph a quadratic equation and analyze the graph to find solutions.īefore we get started, let’s review a few things. Hello, and welcome to this video about solutions of a quadratic on a graph! Today we’ll learn how to find solutions to a quadratic function by looking at its graph.
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