![]() ![]() The maximum or minimum point of a quadratic is called the vertex. The points have the form (t, h).Īccording to the graph, the rock reaches its greatest height at 2 seconds. The height of the rock depends on the time, so h is the dependent variable, and t is the independent variable. Graph the points obtained in parts a through e. The rock is zero feet in the air at 4 seconds that is, the rock has hit the ground.į. In the formula, h = -16t 2 +64t, replace t with 4. The rock is 48 feet in the air at 3 seconds.Į.ğind the height of the rock when t = 4. In the formula, h = -16t 2 + 64t, replace t with 3. The rock is 64 feet in the air at 2 seconds.Įxplanation: Order of operations requires that you apply exponents before multiplying.ĭ.ğind the height of the rock when t = 3. ![]() In the formula, h = -16t 2 + 64t, replace t with 2. the -16 is multiplied by 1 2Ĭ.ğind the height of the rock when t = 2. The rock is 48 feet in the air at one second.Įxplanation: Only the "1" is being squared. In the formula, h = -16t 2 + 64t, replace t with 1. (This is the point right before he shoots the rock in the air.)ī.ğind the height of the rock when t = 1. The rock is zero feet in the air at zero seconds. In the formula, h = -16t 2 + 64t, replace t with 0. The quadratic equation that models the height of the rock isĪ.ğind the height of the rock when t = 0. A boy lying on his back uses a sling shot to fire a rock straight up in the air with an initial velocity (the force the boy uses to fire the rock) of 64 feet per second. This feature of quadratics makes them good models for describing the path of an object in the air or describing the profit of a company (examples of which you may see in Finite Mathematics or in Microeconomics.)Įxample 1. Why study quadratics? The graphs of quadratic equations result in parabolas (U shaped graphs that open up or down). The slope-intercept equation from the second chapter, y = mx + b is called a first degree polynomial because the highest exponent is one. Quadratics are also called second degree polynomials because the highest exponent is 2. Vocabulary: The standard format of a quadratic equation is y = ax 2 + bx + c a, b, c are constants x is the independent variable, and y is the dependent variable. In this section, you will add, subtract, multiply, and graph quadratics. (Note: students will likely need to experiment quite a bit to find an equation that satisfies these constraints.Chapter 4 - QUADRATICS INTRODUCTION TO QUADRATICS Objectives The following go through the points $(-4,2)$ and $(1, 2)$: The following have $x$-intercepts at the origin and $(-4,0)$: The following have a $y$-intercept of $(0,-6)$ : The following have a vertex at $(-2,-5)$ : Asking students for three possible answers is a great extension for students - it gets them thinking about the effects of the different parts of the equation. We’re including three possible answers for each one, to demonstrate the type of variability you might expect to see in a class. Note: each of these problems has many possible answers. The $x$-intercepts are $(3,0)$ and $(-1, 0)$, which are most visible in $y_1$ since you can find the roots of the polynomial using the zerofactor property and thus the intercepts correspond to the zeros of each factor. The $y$-intercept is $(0, -3)$, which is visible as the constant in $y_2$ since the other terms are 0 when you plug in $x = 0$. The vertex is $(1, -4)$ which is most visible in $y_3$ since the vertex occurs at the point where the squared portion is zero. We can see that the difference between it and $y_2$ is just 4, so that graph is 4 units below the other one. The fourth function produces a different graph. Similarly, if we multiply out the perfect square and combine like terms in the third equation, we also get the second one: If we multiply the factors given in the first equation, we’ll get the second equation: ![]() This is because the first three equations are equivalent, and so all produce the same graph. When you graph these four equations, only two different parabolas are shown. ![]()
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