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## Quadratic Equations

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**Quadratic Equations**Function of a different shape**There are many uses of parabolas in real-world applications.****Graphs of Quadratic Functions**Plotting quadratic curves**If you remember a relation is a correspondence between two**sets of numbers called the domain and range. If each member of the domain is assigned exactly one member of the range, then the relation is a function.**A function can be represented as a list or a table of**ordered pairs, a graph in the coordinate plane, or an equation in two variables.**If you notice, the right side of the equationy = 3x + 2 is a**polynomial. Can you classify the polynomial by degree?A function of this form (y = mx + b) is called a linear function. Note the graph is a straight line. Y = 3x + 2**Now consider the equationy = x2 + 6x – 1Classify the**polynomial on the right.A function defined by an equation of this form y = ax2 + bx + c is called a quadraticfunction.Now we are going to investigate this form.**Let’s try a little experiment with your graphing**calculator.Graph the equation y = x2on the coordinate plane. Now graph y = 3x2on the same coordinate plane.How are the graphs the same?How are they different?Can you predict how the graph of y = ¼x2will be similar or different? HINT: Graph y = x2 first, next graph y = 3x2 to see what differences or similarities are present. Now graph y = ¼x2 to see how the shape of the graph changes.**Type in y = x2 on the graphing calculator.**Graph of y = x2 Now graph y = 3x2**Type in y = 3x2 on the graphing calculator.**Graph of y = x2 and y = 3x2 on the same graph. How are the graphs the same? How are they different? Can you predict what y = ¼x2 looks?**Type in y = x2, y = 3x2, and y = ¼x2.**Graphs of y = x2, y = 3x2, and y = ¼x2. What differences do you notice in the new graph y = ¼x2.**These functions (equations) are quadratic functions.**Standard Form of a Quadratic Function A quadratic function - is a function that can be written in the form y = ax2 + bx + c, where a ≠ 0. This form is called the standard form of a quadratic function. Ex: y = 5x2 y = x2 + 7 y = x2 –x -3**The variable in a quadratic function is squared (x2), so the**graph forms a curved line called a parabola. All quadratic functions have the same shape. The graph of y = x2 forms this U-shaped graph called a parabola.**You can fold a parabola so that the two sides match exactly.**This property is called symmetry. The fold or line that divides the parabola into two matching halves is called the axis of symmetry. Axis of symmetry**The highest or lowest point of a parabola is its vertex,**which is on the axis of symmetry.If a > 0 in (a positive number) y = ax2 + bx + cthe parabola opens upward. The vertex is the minimumpoint or lowest point of the graph.If a < 0 in (a negative number)y = -ax2 + bx + cthe parabola opens downward. The vertex is the maximumpoint or highest point of the graph Minimum point Maximum point**Identifying a VertexIdentify the vertex of each graph. Tell**whether it is a minimum or maximum.**On the left is the graph of a parabola. Below are examples**of equations of parabolas.y = x2x = y2y = x2 + 2x + 3**You can use the fact that a parabola is symmetric to graph**it quickly.First, find the coordinates of the vertex and several points on either side of the vertex. Then reflect the points across the axis of symmetry. For functions of the form of y = ax2, the vertex is at the origin.Make a table for the function y = x2using x = 0, 1, 2, and 4. Remember our original experiment when we graphed y = x2 and y = 3x2 Try this**Make a table for y = x2using x = 0, 1, 2, 4.Graph the points**on the graph then reflect the x-values to the other side of the graph. Try another**Make a table of values and graph the quadratic function f(x)**= -2x2using x = 0, 1, 2, 4 Remember these are functions so we also use the function notationf(x)**Graphing y = ax2 + c (y = 2x2 + 4)The value of c, the**constant term in a quadratic function, translates the graph up or down. Make a table and graph y = 2x2 Make a table and graph Y = 2x2 + 3 Try it**Graph y = x2 and y = x2 - 4**This time let’s use the same graph for both. Real world application**Real World ProblemYou can model the height of an object**moving under the influence of gravity using a quadratic function. As an object falls, its speed continues to increase. You can find the height of a falling object using the function h = -16t2 + c.The height h is in feet, the time t is in seconds, and the initial height of the object c is in feet. The graph shows at 0 seconds the object is at 50 feet, after one second the object has already fallen to 34 feet, and at 1½ seconds the object has hit the ground height (feet) seconds Seagull drops a clam to break the shell so it can eat it. The gull drops the clam from 50 feet in the air. Try one**Suppose a squirrel is in a tree 60 feet off the ground. She**drops an acorn. The functionh = -16t2 + 60 gives the height h of the acorn in feet after t seconds. Make a table and graph this function.**Graph each function**• y = -x2 • y = 2x2 • y = 3x2 – 6 • y = -½x2 + 3 Match the graph**Can you match these graphs with their functions?f(x) = x2 +**4 f(x) = -x2 + 2**Graph of a Quadratic Function**f(x) = ax2 + bx + c**So far we have investigated the graphs of y = ax2and y = ax2**+ c. In these functions c has always been 0, which means the axis of symmetry has always been the y-axis.In the quadratic function y = ax2 + bx + c, thevalue of baffects the position of the axis of symmetry, moving it left or right.In the next slide we are going to consider functions in the form of y = ax2 + bx + c**Notice that both graphs have the same y-intercept. This is**because in both equations c = 0 y = 2x2 + 2x Y = 2x2 + 4x The axis of symmetry changes with each change in the b value.**Since the axis of symmetry is related to the change in the b**value, the equation of the axisof symmetry is related to the ratio b/a x = -b/2a Let’s try one! To find the y-value, first substitute a and b into theequationx = -b/2a and solve to find x. Then substitute x back into the original equation to determine y.**Find the coordinates of the vertex and an equation for the**axis of symmetry. Then graph the function.y = x2 – 4x + 3a = 1, b = -4, and c = 3x = -b/2ax = -(-4)/2(1) = 4/2 = 2axis of symmetry: x = 2If x = 2,then y = x2 -4x + 3 = y = 22 - 4(2) + 3 = -1 Use the equation for the axis of symmetry. x = -b/2a Substitute the x-value into the original equation and solve for y. the vertex is (2, -1) Now make a table.**Since the vertex of the axis of symmetry is (2, -1) and we**know the parabola turns upward (a > 0), we can use values on both sides of (2, -1). Now graph your points and draw a curved line. Try this one.**y = -x2 + 4**a = ,b = c = Find x = -b/2a x = Substitute the x-value into the equation y = -x2 + 4 Now use your vertex as the middle of your table. Solve problems**Graph each function. Label the axis of symmetry and the**vertex.1) y = x2 + 4x + 32) y = 2x2 – 6x3) y =x2 + 4x – 44) y = 2x2 + 3x + 1 Real world problem**In professional fireworks displays, aerial fireworks carry**“stars” upward, ignite them, and project them into the air.The equation h = -16t2 + 72t + 520 gives the star’s height h in feet at time t in seconds. Since the coefficient of t2 is negative, the curve opens downward, and the vertex is the maximum point.Find the t-coordinate of the vertexx = -b/2a = -72/2(-16) = 2.25After 2.25 seconds, the star will be at its greatest height.Find the h-coordinate of the vertex.h = -16(2.25)2 + 72(2.25) + 520 = 601The maximum height of the star will be 601 feet. TRY THIS**The shape of the Gateway Arch in St. Louis, Missouri, is a**catenary curve that resembles a parabola. The equationh = -0.00635x2 + 4.0005x – 0.07875 represents the parabola, where h is the height in feet and x is the distance from one base in feet.What is the equation of the axis of symmetry?What is the maximum height of the arch?**Using the Quadratic Formula**Solving any quadratic equations.**In our earlier lesson, you solved quadratic equations by**factoring. Another method, which will solve any quadratic equation, is to use the quadratic formula as seen left.Here values of a, b, and c are substituted into the formula to determine x.**Be sure to write a quadratic equation in standard form**before using the quadratic formula.Solve: x2 + 6 = 5xx2 -5x + 6 = 0**You can use the quadratic formula to solve real-world**problems. Suppose a football player kicks a ball and gives it an initial velocity of 47ft/s. The starting height of the football is 3 ft. If no one catches the football how long will it be in the air?Using the vertical motion formula and the information given, the formulah = -16t2 + vt + crepresents this illustration. VERTICAL MOTION FORMULA h = -16t2 + vt + c The initial upward velocity is v, and the starting height is c You must decide whether a solution makes sense in the real-world situation. For example, a negative value for time is not a reasonable solution.**Use the vertical motion formula h = -16t2 + vt + c1) A**child tosses a ball upward with a starting velocity of 10 ft/s from a height of 3 ft.a. Substitute the values into the vertical motion formula. Let h = 0b. Solve. If it is not caught, how long will the ball be in the air? Round to the nearest tenth of a second.2) A soccer ball is kicked with a starting velocity of 50 ft/s from a starting height of 3.5 ft.a. Substitute the values into the vertical motion formula. Let h = 0b. Solve. If no one touches the ball, how long will the ball be in the air? TRY THIS**The function below models the United States population P in**millions since 1900, where t is the number of years after 1900.P = 0.0089t2 + 1.1149t + 78.4491a. Use the function to estimate the US population the year I graduated from high school.b. Estimate the US population in 2025.c. Estimate the US population in 2050. Try Another**A carnival game involves striking a lever that forces a**weight up a tube to strike a bell which will win you a prize. If the weight reaches 20 feet and strikes the bell, you win. The equationh = -16t2 + 32t + 3gives the height h of the weight if the initial velocity v is 32 ft/s.Find the maximum height of the weight.Will the contestant win a prize? One More**The Sky Concert in Peoria, Illinois, is a 4th of July**fireworks display set to music. If a rocket (firework) is launched with an initial velocity of 39.2 m/s at a height of 1.6 m above the ground, the equation,h = -4.9t2 + 39.2t + 1.6represents the rockets height h in meters after t seconds. The rocket will explode at approximately the highest point.At what height will the rocket explode?**ReviewIf a quadratic equation is written in the form ax2 +**bx + c = 0, the solutions can be found using the quadratic formula.In the quadratic equation, the expression under the radical sign, b2 – 4ac, is called the discriminant.1) If b2 – 4ac is a negative number, the square root cannot be found as a real number. There are no real-number solutions.2) If b2 – 4ac equals 0, there is only one solution of the equation.3) If b2 – 4ac is a positive number, there are two solutions of the equation. The graph of the quadratic intersects the x-axis twice.

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