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Algebra I Honors—Semester 2Extension Activity 1 of 2ObjectivesAnalyze features of parabolas and their graphs. Identify the characteristics of the three types of systems of equations.Explain the methods of solving systems of equations. Solve systems of equations. Submitting AssignmentsOption 1—Type answers in the space provided using a mathematical writing program and email your work through Stars Suite email (using copy/paste). Option 2—Print this document, complete your work by hand, scan it, and email it to your online instructor through normal email at his or her Academy email address.AssignmentTopic—Parabolas Answer the questions using the vertex form of a parabolic function. The standard form for a parabola is y=a(x-h)2+k.What do the values of a, h, and k represent?The value of a represents the size and direction of the parabola. The larger the absolute value of a, the narrower the parabola is. The value of h represents the x-coordinate of the vertex of the parabola, and the value of k represents the y-coordinate of the vertex of the parabola.What happens to the shape of the parabola y= ax2 when a increases from 0? (For example, what happens to the graphs when a is 1 and then when a is 15)?As the value of a increases from 0, the shape of the parabola y=ax2 gets narrower. So, the graph of y=15x2 will be a narrower parabola than the graph of y=x2.Describe the shape of the parabola y= ax2 when a is negative.When a is negative, the parabola y=ax2 will open downwards.What happens to the graph of the parabola y=x2+k when k is positive?When k is positive, the graph of the parabola is shifted vertically upward k units.What happens to the graph of the parabola y=x2+k when k is negative?When k is negative, the graph of the parabola y= x2+k is shifted vertically downward k units.What happens to the graph of the parabola y=(x-h)2 when h changes from positive to negative?When h changes from a positive to a negative value, the vertex of the parabola y=(x-h)2 shifts from being to the right of the y-axis to being to the left of the y-axis. So, the x-value of the vertex changes from negative to ic—Systems of EquationsWhat is the meaning of a solution to a system of equations?Student answers may vary. A solution to a system of equations is an ordered pair that will satisfy all the equations in the system. So, if an ordered pair is a solution, it will make all the equations in the system true.A system of linear equations can be classified in one of three ways. The lines may intersect at one unique point, the lines may be parallel and unique, or the lines may coincide. Use the words in the table to describe the slope, y-intercepts, and number of solutions in each type of system of equations. (Words can be used more than once, and some answers may require more than one word).Slopey-interceptNumber of solutionsChoicesSameDifferentSame DifferentOneNoneInfiniteOne Unique point: Different slope(s), same or different y-intercept(s), one solution(s)Parallel: Same slope(s), different y-intercept(s), none solution(s)Coincide: Same slope(s), same y-intercept(s), infinite solution(s)When you use the linear combination method, how can you tell whether you have to multiply by a constant first?Student answers may vary. When using the linear combination method, you have to multiply by a constant when neither the x- nor the y- values have opposite coefficients.Explain why you want to eliminate a variable when you substitute or combine equations in a system.Student answers may vary. You want to eliminate a variable when you substitute or combine equations in a system in order to get one equation that has only one variable. Then, instead of having a system of equations to solve, you will just have one equation with one unknown variable to solve for.Solve each system of equations. Show your work and final solution. 81=-10x+9y -9y-4x-45=0Start by eliminating the y-terms, then solve for x. -10x+9y= 81+ (-4x-9y= 45)25711152540 -14x=126 x=-9Next, substitute x=-9 into one of the original equations and solve for y.81= -10-9+9y-9=9y-1=yTherefore, the final solution is (-9,-1).The graphs of ax-by=17 and ax+by=-1 intersect at 4, 3. Find a and b. The point (4,3) is a solution to both equations, so start by substituting 4 for x and 3 for y. Then eliminate the b-terms and solve for a. 4a-3b=17+ 4a+3b= -12371725635 8a=16 a=2Next, substitute a = 2 into one of the equations and solve for b. 42-3b=17-3b=9b=-3 ................
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