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17 (a) Write $x^2 + 8x + 3$ in the form $(x + a)^2 - b$ - OCR - GCSE Maths - Question 18 - 2019 - Paper 5
Question 18
17 (a) Write $x^2 + 8x + 3$ in the form $(x + a)^2 - b$. (b) Sketch the graph of $y = x^2 + 8x + 3$. Show clearly the coordinates of any turning points and the y-in... show full transcript
Worked Solution & Example Answer:17 (a) Write $x^2 + 8x + 3$ in the form $(x + a)^2 - b$ - OCR - GCSE Maths - Question 18 - 2019 - Paper 5
Step 1
Write $x^2 + 8x + 3$ in the form $(x + a)^2 - b$.
Answer
To rewrite the quadratic expression in the required form, we can complete the square. Start with the expression:
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Take the coefficient of , which is , divide it by , and square it:
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Now rewrite the expression as:
This simplifies to:
Thus, we have:
Step 2
Sketch the graph of $y = x^2 + 8x + 3$. Show clearly the coordinates of any turning points and the y-intercept.
Answer
To sketch the graph of the quadratic function , we can identify its vertex and y-intercept.
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Turning Point (Vertex): We have already rewritten the function as:
The vertex is at the point .
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Y-Intercept: To find the y-intercept, we set :
Therefore, the y-intercept is at .
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Graphing: Plot the vertex at and the y-intercept at . Since the parabola opens upwards (the coefficient of is positive), sketch the curve accordingly, ensuring the points are labeled. The graph should reflect the shape of a parabola opening upwards, with the turning point as the minimum point.