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Questions and Answers
What is the equation of the axis of symmetry for the function $y = -x^2 + x + 7$?
What is the equation of the axis of symmetry for the function $y = -x^2 + x + 7$?
What is the minimum value of the function $f(x) = x^2 - 3x - 3$?
What is the minimum value of the function $f(x) = x^2 - 3x - 3$?
When $x = 2$, what does the function $f(x) = x^2 -3x - 3$ evaluate to?
When $x = 2$, what does the function $f(x) = x^2 -3x - 3$ evaluate to?
For the function $y = x^2$, at what value of $x$ does the y-coordinate equal 9?
For the function $y = x^2$, at what value of $x$ does the y-coordinate equal 9?
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What coordinates correspond to the maximum point of the function $y = -x^2 + x + 7$?
What coordinates correspond to the maximum point of the function $y = -x^2 + x + 7$?
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What is the solution for $x^2 - 3x - 3 = 5$?
What is the solution for $x^2 - 3x - 3 = 5$?
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What is the gradient of the line MN given the points $(-1, y)$ and $(x, 9)$?
What is the gradient of the line MN given the points $(-1, y)$ and $(x, 9)$?
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Which of the following represents the correct equation of the line parallel to MN passing through the origin?
Which of the following represents the correct equation of the line parallel to MN passing through the origin?
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Study Notes
Determining Equation of Lines
- Two lines g and h are perpendicular
- Line g passes through points (0,5) and (4,1)
- Slope of g is (1-5) / (4-0) = -1
- Equation of g is y = -x + 5
- Line h passes through (2,3)
- Using the formula for perpendicular lines, the slope of h is 1
- Equation of h is y - 3 = 1 (x - 2)
- Simplify h to y = x + 1
- Point P is the intersection between g and h
- Substitute y = -x + 5 into y = x + 1
- This gives -x +5 = x + 1
- Solve the equation to find x = 2
- Substitute x = 2 into y = x + 1
- This gives y = 3
- Coordinates of point P are (2, 3)
Completing the Square
- Express 4x² - 24x + 31 in the form a(x + h)² + k
- Step 1: Factor out the coefficient of x² which is 4
- Step 2: Complete the square within the parenthesis: Take half of the coefficient of the x term (-6), square it (36) and add and subtract it inside the parenthesis to maintain equality.
- Step 3: Simplify, move the constant term outside the parenthesis and multiply it by 4
- 4x² - 24x + 31 becomes 4(x - 3)² - 1
- The graph is a parabola, since a is positive, the parabola opens upwards
- The vertex of the parabola is (3, -1)
- The y-intercept is found by setting x to 0 in the equation, this results in y = 31
- For the axis of symmetry, the equation is x = 3
Analyzing a Quadratic Function
- The graph represents the function f(x) = x² -3x - 3
- When x = 2, f(x) is -1
- When x = 1.5, f(x) is - 4.25
- The values of x for which f(x) = 0 are approximately 3.8 and -0.8
- The minimum value of f(x) is -4.25
- The minimum value of f(x) occurs at x = 1.5
- The solution for x² - 3x - 3 = 5 is found by identifying where the graph intersects the line y = 5
- The solutions are approximately -1.2 and 4.2
- The interval on the domain for which f(x) is less than -3 is x < 1.5 and x > 3.8
Determining Values and Equations
- The point M is (-1, y) and N is (x, 9)
- y is found by substituting x = -1 in the equation y = x²
- y = 1
- x is found by substituting y = 9 in the equation y = x²
- x = 3
Calculating Gradient and Equation of a Line
- The gradient of line MN is calculated using the formula (9 - 1)/(3 - (-1)) = 2
- The equation of line MN is y - 1 = 2(x + 1)
- Simplify the equation to y = 2x + 3
- The equation of the line parallel to line MN passing through the origin is y = 2x
Graphing a Quadratic Function
- Table of values for the function y = -x² + x + 7 is to be completed
- When x = 2, y is -1
- When x = 3, y is -5
- When x = 4, y is -9
- Using the given scale, the graph of the function is to be drawn
- The maximum point of the graph is (0.5, 7.25)
- The equation of the axis of symmetry is x =0.5
- The solutions to the equation -x² + x + 7 = 0 are the x-intercepts of the graph, approximately -2.5 and 3
- The equation of the line passing through the points (-3, -1) and (0, 8) can be determined as follows:
- Find the slope m: m = (8 - (-1)) / (0 - (-3)) = 3
- Find the y-intercept c: Substitute the point (0, 8) into the equation y = mx + c and solve for c. This results in c = 8.
- The equation of the line is y = 3x + 8
Analyzing a Graph
- Values of x and y for the function y = x² - 2x - 3 are to be calculated for integer values of x from -2 to 4
- These values should be recorded in the table provided.
- Based on the provided table, a graph representing the function should be drawn
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Description
This quiz focuses on determining the equations of perpendicular lines and completing the square for quadratic expressions. You will analyze two lines, find their intersection point, and practice transforming quadratic equations into vertex form. Test your understanding of these key algebra concepts.
