name:
week 15 quiz
answer the following questions regarding quadratics in vertex form $y = a(x - h)^2 + k$
1. what makes the graph wider and narrower?
2. what makes the graph move left and right?
3. what makes the graph move up and down?
for the following, find a) the axis of symmetry, b) the vertex, and c) the y intercept.
4. $y = x^2 - 2x + 1$
5. $y = x^2 - 4x + 3$
6. $y = x^2 - 6x + 5$
7. $y = -2x^2 - 8x + 10$
factor
8. $x^2 - 7x - 8$
9. $x^2 - 5x + 6$
10. $x^2 + 4x - 12$
11. $x^2 + 6x + 5$

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Accepted Answer

### Question 1: What makes the graph wider and narrower? # Brief Explanations: In the vertex form of a quadratic function $ y = a(x - h)^2 + k $, the coefficient $ a $ determines the width and direction of the parabola. If $ |a|>1 $, the graph is narrower than the standard parabola $ y = x^2 $. If $ 0<|a|<1 $, the graph is wider than the standard parabola $ y = x^2 $. So the value of $ a $ (the coefficient of the squared term) in the vertex form determines if the graph is wider (when $ 0 < |a| < 1 $) or narrower (when $ |a|>1 $). # Answer: The value of $ a $ in the vertex form $ y=a(x - h)^2 + k $. If $ 0<|a|<1 $, the graph is wider; if $ |a| > 1 $, the graph is narrower. ### Question 2: What makes the graph move left and right? # Brief Explanations: In the vertex form $ y=a(x - h)^2 + k $, the value of $ h $ is responsible for the horizontal shift (left or right) of the parabola. The vertex of the parabola is at $ (h,k) $. If we replace $ h $ with $ h + c $ (where $ c>0 $), the graph shifts to the left by $ c $ units, and if we replace $ h $ with $ h - c $ (where $ c>0 $), the graph shifts to the right by $ c $ units. So the value of $ h $ in the vertex form determines the horizontal (left - right) movement of the graph. # Answer: The value of $ h $ in the vertex form $ y=a(x - h)^2 + k $. If $ h $ decreases (for a given $ a $ and $ k $), the graph moves left; if $ h $ increases, the graph moves right. ### Question 3: What makes the graph move up and down? # Brief Explanations: In the vertex form $ y=a(x - h)^2 + k $, the value of $ k $ is responsible for the vertical shift (up or down) of the parabola. The vertex of the parabola is at $ (h,k) $. If we increase the value of $ k $, the vertex moves up, and thus the entire graph of the parabola moves up. If we decrease the value of $ k $, the vertex moves down, and thus the entire graph of the parabola moves down. So the value of $ k $ in the vertex form determines the vertical (up - down) movement of the graph. # Answer: The value of $ k $ in the vertex form $ y=a(x - h)^2 + k $. If $ k $ increases, the graph moves up; if $ k $ decreases, the graph moves down. ### Question 4: For $ y=x^{2}-2x + 1 $ #### a) Axis of symmetry # Explanation: ## Step 1: Recall the formula for axis of symmetry For a quadratic function in the form $ y=ax^{2}+bx + c $, the axis of symmetry is given by $ x=-\frac{b}{2a} $. For the function $ y=x^{2}-2x + 1 $, we have $ a = 1 $, $ b=-2 $, and $ c = 1 $. ## Step 2: Calculate the axis of symmetry Using the formula $ x=-\frac{b}{2a} $, substitute $ a = 1 $ and $ b=-2 $. We get $ x=-\frac{-2}{2\times1}=\frac{2}{2}=1 $. # Answer (a): The axis of symmetry is $ x = 1 $ #### b) Vertex # Explanation: ## Step 1: Recall the vertex coordinates The vertex of a quadratic function $ y=ax^{2}+bx + c $ lies on the axis of symmetry. So the x - coordinate of the vertex is the same as the axis of symmetry, which we found to be $ x = 1 $. ## Step 2: Find the y - coordinate of the vertex Substitute $ x = 1 $ into the function $ y=x^{2}-2x + 1 $. We get $ y=(1)^{2}-2\times1 + 1=1-2 + 1=0 $. So the vertex is $ (1,0) $. # Answer (b): The vertex is $ (1,0) $ #### c) y - intercept # Explanation: ## Step 1: Recall the definition of y - intercept The y - intercept of a function is the value of $ y $ when $ x = 0 $. ## Step 2: Calculate the y - intercept Substitute $ x = 0 $ into the function $ y=x^{2}-2x + 1 $. We get $ y=0^{2}-2\times0+1 = 1 $. # Answer (c): The y - intercept is $ y = 1 $ (or the point $ (0,1) $) ### Question 5: For $ y=x^{2}-4x + 3 $ #### a) Axis of symmetry # Explanation: ## Step 1: Identify $ a $ and $ b $ For the quadratic function $ y=x^{2}-4x + 3 $, $ a = 1 $ and $ b=-4 $. ## Step 2: Use the axis of symmetry formula Using the formula $ x=-\frac{b}{2a} $, substitute $ a = 1 $ and $ b=-4 $. We get $ x=-\frac{-4}{2\times1}=\frac{4}{2}=2 $. # Answer (a): The axis of symmetry is $ x = 2 $ #### b) Vertex # Explanation: ## Step 1: Find the x - coordinate of the vertex The x - coordinate of the vertex is the same as the axis of symmetry, so $ x = 2 $. ## Step 2: Find the y - coordinate of the vertex Substitute $ x = 2 $ into the function $ y=x^{2}-4x + 3 $. We get $ y=(2)^{2}-4\times2+3=4 - 8 + 3=-1 $. So the vertex is $ (2,-1) $. # Answer (b): The vertex is $ (2,-1) $ #### c) y - intercept # Explanation: ## Step 1: Substitute $ x = 0 $ Substitute $ x = 0 $ into the function $ y=x^{2}-4x + 3 $. ## Step 2: Calculate $ y $ We get $ y=0^{2}-4\times0 + 3=3 $. # Answer (c): The y - intercept is $ y = 3 $ (or the point $ (0,3) $) ### Question 6: For $ y=x^{2}-6x + 5 $ #### a) Axis of symmetry # Explanation: ## Step 1: Identify $ a $ and $ b $ For the quadratic function $ y=x^{2}-6x + 5 $, $ a = 1 $ and $ b=-6 $. ## Step 2: Use the axis of symmetry formula Using the formula $ x=-\frac{b}{2a} $, substitute $ a = 1 $ and $ b=-6 $. We get $ x=-\frac{-6}{2\times1}=\frac{6}{2}=3 $. # Answer (a): The axis of symmetry is $ x = 3 $ #### b) Vertex # Explanation: ## Step 1: Find the x - coordinate of the vertex The x - coordinate of the vertex is $ x = 3 $ (from the axis of symmetry). ## Step 2: Find the y - coordinate of the vertex Substitute $ x = 3 $ into the function $ y=x^{2}-6x + 5 $. We get $ y=(3)^{2}-6\times3+5=9 - 18 + 5=-4 $. So the vertex is $ (3,-4) $. # Answer (b): The vertex is $ (3,-4) $ #### c) y - intercept # Explanation: ## Step 1: Substitute $ x = 0 $ Substitute $ x = 0 $ into the function $ y=x^{2}-6x + 5 $. ## Step 2: Calculate $ y $ We get $ y=0^{2}-6\times0 + 5=5 $. # Answer (c): The y - intercept is $ y = 5 $ (or the point $ (0,5) $) ### Question 7: For $ y=-2x^{2}-8x + 10 $ #### a) Axis of symmetry # Explanation: ## Step 1: Identify $ a $ and $ b $ For the quadratic function $ y=-2x^{2}-8x + 10 $, $ a=-2 $ and $ b=-8 $. ## Step 2: Use the axis of symmetry formula Using the formula $ x=-\frac{b}{2a} $, substitute $ a=-2 $ and $ b=-8 $. We get $ x=-\frac{-8}{2\times(-2)}=\frac{8}{-4}=-2 $. # Answer (a): The axis of symmetry is $ x=-2 $ #### b) Vertex # Explanation: ## Step 1: Find the x - coordinate of the vertex The x - coordinate of the vertex is $ x=-2 $ (from the axis of symmetry). ## Step 2: Find the y - coordinate of the vertex Substitute $ x=-2 $ into the function $ y=-2x^{2}-8x + 10 $. We get $ y=-2\times(-2)^{2}-8\times(-2)+10=-2\times4 + 16 + 10=-8 + 16+10 = 18 $. So the vertex is $ (-2,18) $. # Answer (b): The vertex is $ (-2,18) $ #### c) y - intercept # Explanation: ## Step 1: Substitute $ x = 0 $ Substitute $ x = 0 $ into the function $ y=-2x^{2}-8x + 10 $. ## Step 2: Calculate $ y $ We get $ y=-2\times0^{2}-8\times0 + 10=10 $. # Answer (c): The y - intercept is $ y = 10 $ (or the point $ (0,10) $) ### Question 8: Factor $ x^{2}-7x - 8 $ # Explanation: ## Step 1: Recall the factoring form We need to find two numbers that multiply to $ -8 $ (the constant term) and add up to $ -7 $ (the coefficient of the middle term). ## Step 2: Find the two numbers The numbers are $ -8 $ and $ 1 $ because $ -8\times1=-8 $ and $ -8 + 1=-7 $. ## Step 3: Write the factored form So $ x^{2}-7x - 8=(x - 8)(x+1) $. # Answer: $ (x - 8)(x + 1) $ ### Question 9: Factor $ x^{2}-5x + 6 $ # Explanation: ## Step 1: Recall the factoring form We need to find two numbers that multiply to $ 6 $ (the constant term) and add up to $ -5 $ (the coefficient of the middle term). ## Step 2: Find the two numbers The numbers are $ -2 $ and $ -3 $ because $ (-2)\times(-3)=6 $ and $ -2+(-3)=-5 $. ## Step 3: Write the factored form So $ x^{2}-5x + 6=(x - 2)(x - 3) $. # Answer: $ (x - 2)(x - 3) $ ### Question 10: Factor $ x^{2}+4x - 12 $ # Explanation: ## Step 1: Recall the factoring form We need to find two numbers that multiply to $ -12 $ (the constant term) and add up to $ 4 $ (the coefficient of the middle term). ## Step 2: Find the two numbers The numbers are $ 6 $ and $ -2 $ because $ 6\times(-2)=-12 $ and $ 6+(-2)=4 $. ## Step 3: Write the factored form So $ x^{2}+4x - 12=(x + 6)(x - 2) $. # Answer: $ (x + 6)(x - 2) $ ### Question 11: Factor $ x^{2}+6x + 5 $ # Explanation: ## Step 1: Recall the factoring form We need to find two numbers that multiply to $ 5 $ (the constant term) and add up to $ 6 $ (the coefficient of the middle term). ## Step 2: Find the two numbers The numbers are $ 1 $ and $ 5 $ because $ 1\times5 = 5 $ and $ 1 + 5=6 $. ## Step 3: Write the factored form So $ x^{2}+6x + 5=(x + 1)(x + 5) $. # Answer: $ (x + 1)(x + 5) $

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Question 1: What makes the graph wider and narrower?

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Question 2: What makes the graph move left and right?