name aya miai 7.1 - 7.2 quiz multiple choice identify the choice that best completes the statement or answers the question. 1. which description of a graph appears to represent a quadratic relation? a. a line straight up and down b. an ellipse c. a parabola opening up d. a parabola opening to the right 2. which relation is quadratic? a. y = x³ - x² + 4x + 2 b. y = (2x²)(x + 1) c. y = (x + 5)² d. y = 2x - 6x + 3 3. what is the degree of a quadratic function? a. 3 b. 2 c. 0 d. 1 4. what is the y-intercept for y = 3x² + 2x - 5? a. -5 b. 5 c. 2 d. 3 5. which parabola opens upward? a. y = 2x - 4x² - 5 b. y = 2 + 4x - 5x² c. y = 4 - 2x² - 5x d. y = -5x + 4x² + 2 6. the points (-2, 4) and (1, 4) are located on the same parabola. what is the equation for the axis of symmetry for this parabola? a. x = -0.5 b. x = -1 c. x = 0.5 d. x = -1.5 7. which set of data is correct for this graph? (graph of a parabola) set axis of symmetry vertex domain range a. y = -2 (-2, 6) x ∈ r y ∈ r b. x = -6 (-6, -2) -8 ≤ x ≤ 4 -8 ≤ y c. x = -2 (-2, -6) x ∈ r -6 ≤ y d. x = 2 (2, 6) -6 ≤ x ≤ 2 -6 ≤ y a. set a. b. set b. c. set d. d. set c. 8. each table of values lists points in a quadratic relation. decide without graphing, the direction in which the parabola opens. a) opens and b) opens a) x -4 -3 -2 -1 0 1 y 12 5 0 -3 -4 -3 b) x 0 1 2 3 4 5 y -13 -3 3 5 3 -3

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### Question 1 # Explanation: ## Step1: Recall quadratic relation graph A quadratic relation's graph is a parabola. A vertical line (a) is linear, an ellipse (b) is conic but not quadratic relation's typical graph, a parabola opening right (d) is a horizontal parabola (quadratic in $y$), but the standard quadratic relation ($y = ax^2+bx + c$) has a vertical parabola (opening up/down). So a parabola opening up (c) is a quadratic relation's graph. ## Step2: Eliminate other options - Option a: Vertical line is linear, not quadratic. - Option b: Ellipse is not a quadratic relation's graph (quadratic relation is $y = ax^2+bx + c$ or $x = ay^2+by + c$, ellipse is $Ax^2 + By^2+...=0$ with $A,B$ non - zero and same sign). - Option d: Parabola opening right is quadratic in $y$, but the question is about the description that appears to represent a quadratic relation (most common is vertical parabola). So the correct option is c. # Answer: c. a parabola opening up ### Question 2 # Explanation: ## Step1: Recall quadratic relation definition A quadratic relation is a polynomial of degree 2 in $x$ (or $y$). The general form is $y=ax^{2}+bx + c,a eq0$. ## Step2: Analyze each option - Option a: $y=x^{3}-x^{2}+4x + 2$ is a cubic (degree 3) polynomial, not quadratic. - Option b: $y=(2x^{2})(x + 1)=2x^{3}+2x^{2}$ is a cubic polynomial (degree 3), not quadratic. - Option c: $y=(x + 5)^{2}=x^{2}+10x + 25$ is a quadratic (degree 2) polynomial. - Option d: $y = 2x-6x + 3=-4x + 3$ is a linear (degree 1) polynomial. # Answer: c. $y=(x + 5)^{2}$ ### Question 3 # Explanation: ## Step1: Recall quadratic function definition A quadratic function is of the form $y = ax^{2}+bx + c,a eq0$. The degree of a polynomial is the highest power of the variable. In $y=ax^{2}+bx + c$, the highest power of $x$ is 2. ## Step2: Eliminate other options - Option a: Degree 3 is cubic, not quadratic. - Option c: Degree 0 is constant function. - Option d: Degree 1 is linear function. So the degree of a quadratic function is 2. # Answer: b. 2 ### Question 4 # Explanation: ## Step1: Recall y - intercept definition The $y$ - intercept of a function $y = f(x)$ is the value of $y$ when $x = 0$. ## Step2: Substitute $x = 0$ into $y=3x^{2}+2x - 5$ When $x = 0$, $y=3(0)^{2}+2(0)-5=-5$. # Answer: a. - 5 ### Question 5 # Explanation: ## Step1: Recall parabola opening direction For a parabola in the form $y=ax^{2}+bx + c$, if $a>0$, it opens upward; if $a < 0$, it opens downward. We need to rewrite each option in $y=ax^{2}+bx + c$ form and check the coefficient of $x^{2}$. ## Step2: Analyze each option - Option a: $y=2x-4x^{2}-5=-4x^{2}+2x - 5$, $a=-4<0$ (opens downward). - Option b: $y=2 + 4x-5x^{2}=-5x^{2}+4x + 2$, $a=-5<0$ (opens downward). - Option c: $y=4-2x^{2}-5x=-2x^{2}-5x + 4$, $a=-2<0$ (opens downward). - Option d: $y=-5x + 4x^{2}+2=4x^{2}-5x + 2$, $a = 4>0$ (opens upward). # Answer: d. $y=-5x + 4x^{2}+2$ ### Question 6 # Explanation: ## Step1: Recall axis of symmetry formula for parabola For a parabola, if two points $(x_1,y)$ and $(x_2,y)$ are on the parabola (same $y$ - value), the axis of symmetry is the vertical line $x=\frac{x_1 + x_2}{2}$. ## Step2: Calculate axis of symmetry Given points $(-2,4)$ and $(1,4)$, $x_1=-2,x_2 = 1$. Then $x=\frac{-2 + 1}{2}=\frac{-1}{2}=-0.5$. # Answer: a. $x=-0.5$ ### Question 7 # Explanation: ## Step1: Analyze the graph The graph is a parabola opening upward (since it opens towards positive $y$ - direction) with vertex at $(-2,-6)$ (from the graph, the lowest point is at $x=-2,y = - 6$) and axis of symmetry $x=-2$ (vertical line through vertex). ## Step2: Analyze each set - Set A: Axis of symmetry $y=-2$ (horizontal line), incorrect (axis of symmetry for vertical parabola is vertical line $x = h$). - Set B: Axis of symmetry $x=-6$, vertex $(-6,-2)$, domain $-8\leq x\leq4$ (not all real numbers), incorrect. - Set C: Axis of symmetry $x=-2$, vertex $(-2,-6)$, domain $x\in R$ (since parabola extends infinitely left/right), range $y\geq - 6$ (since it opens upward), correct. - Set D: Axis of symmetry $x = 2$, vertex $(2,6)$, incorrect (graph has vertex at $x=-2,y=-6$). # Answer: d. Set C ### Question 8 # Explanation: ## Step1: Recall direction of parabola from table For a quadratic relation $y = ax^{2}+bx + c$, we can check the rate of change of $y$ with respect to $x$. If the $y$ - values first decrease and then increase (or vice - versa), we can find the vertex. If the $y$ - values have a minimum (then increase) the parabola opens upward; if a maximum (then decrease), opens downward. ## Step2: Analyze table a For table a: - When $x=-4,y = 12$; $x=-3,y = 5$; $x=-2,y = 0$; $x=-1,y=-3$; $x = 0,y=-4$; $x = 1,y=-3$. - The $y$ - values decrease from $x=-4$ to $x = 0$ (from 12 to - 4) and then increase from $x = 0$ to $x = 1$ (from - 4 to - 3). So the parabola has a minimum at $x = 0$, so it opens upward. ## Step3: Analyze table b For table b: - When $x = 0,y=-13$; $x = 1,y=-3$; $x = 2,y = 3$; $x = 3,y = 5$; $x = 4,y = 3$; $x = 5,y=-3$. - The $y$ - values increase from $x = 0$ to $x = 3$ (from - 13 to 5) and then decrease from $x = 3$ to $x = 5$ (from 5 to - 3). So the parabola has a maximum at $x = 3$, so it opens downward. # Answer: A) opens upward and B) opens downward

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Question 1

Step1: Recall quadratic relation graph

Step2: Eliminate other options

Step1: Recall quadratic relation definition

Step2: Analyze each option

Step1: Recall quadratic function definition

Step2: Eliminate other options

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Question 2