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19 (a) Write $x^2 - 10x + 22$ in the form $(x - a)^2 - b$ - OCR - GCSE Maths - Question 19 - 2020 - Paper 1
Question 19
19 (a) Write $x^2 - 10x + 22$ in the form $(x - a)^2 - b$. (b) Sketch the graph of $y = x^2 - 10x + 22$. Show clearly the coordinates of any turning points and the... show full transcript
Worked Solution & Example Answer:19 (a) Write $x^2 - 10x + 22$ in the form $(x - a)^2 - b$ - OCR - GCSE Maths - Question 19 - 2020 - Paper 1
Step 1
Write $x^2 - 10x + 22$ in the form $(x - a)^2 - b$
Answer
To rewrite the quadratic expression in the form , we will complete the square.
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Start with the expression:
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Take the coefficient of , which is , halve it to get , and then square it to get . We will now add and subtract this value:
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This can be rearranged as:
Thus, we have:
Step 2
Sketch the graph of $y = x^2 - 10x + 22$
Answer
To sketch the graph of , we will first identify key features such as turning points and the y-intercept.
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Vertex/Turning Point: From our completed square form , we can see that the turning point (vertex) is at . This is where the minimum value of the function occurs.
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Y-Intercept: To find the y-intercept, we set : Thus, the y-intercept is at .
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Graph Characteristics: The graph is a parabola opening upwards since the coefficient of is positive. It will cross the y-axis at and have its minimum point at .
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Sketch: Draw a standard Cartesian plane and plot the points and . Then, sketch the parabola, ensuring that it opens upwards and passes through the y-intercept.