It is given that the graph of quadratic function f(x)=2x²+ kx+7 passes through the point (5,17).
(i) Calculate the value of k.
(ii) Find the equation of the axis of symmetry of the graph of quadratic
function.
(iii) Find the coordinates of the minimum point of the graph of the quadratic
function.

Q: It is given that the graph of quadratic function f(x)=2x²+ kx+7 passes through the point (5,17). (i) Calculate the value of k. (ii) Find the equation of the axis of symmetry of the graph of quadratic function. (iii) Find the coordinates of the minimum point of the graph of the quadratic function.

∻Solution by Steps∻ ‖ step 1 ⋮ To find the value of $ k $, we substitute the point $ (5, 17) $ into the quadratic function $ f(x) = 2x^2 + kx + 7 $: $ 2(5)^2 + k(5) + 7 = 17 $. ‖ ‖ step 2 ⋮ Simplifying the equation: $ 2(25) + 5k + 7 = 17 $ leads to $ 50 + 5k + 7 = 17 $. ‖ ‖ step 3 ⋮ Combining like terms gives $ 5k + 57 = 17 $. Subtracting 57 from both sides results in $ 5k = 17 - 57 $, which simplifies to $ 5k = -40 $. ‖ ‖ step 4 ⋮ Dividing both sides by 5 yields $ k = -8 $. ‖ ‖ step 5 ⋮ The equation of the axis of symmetry for a quadratic function $ ax^2 + bx + c $ is given by $ x = -\frac{b}{2a} $. Here, $ a = 2 $ and $ b = -8 $. ‖ ‖ step 6 ⋮ Substituting the values gives $ x = -\frac{-8}{2(2)} = \frac{8}{4} = 2 $. ‖ ‖ step 7 ⋮ To find the coordinates of the minimum point, we substitute $ x = 2 $ back into the function $ f(x) = 2x^2 + kx + 7 $: $ f(2) = 2(2)^2 + (-8)(2) + 7 $. ‖ ‖ step 8 ⋮ Simplifying gives $ f(2) = 2(4) - 16 + 7 = 8 - 16 + 7 = -1 $. Thus, the minimum point is $ (2, -1) $. ‖ ∻Answer∻ ⚹ (i) $ k = -8 $, (ii) Axis of symmetry: $ x = 2 $, (iii) Minimum point: $ (2, -1) $ ⚹ ∻Key Concept∻ ⚹ The axis of symmetry and minimum point of a quadratic function are derived from its coefficients. ⚹ ∻Explanation∻ ⚹ The value of $ k $ was found by substituting the given point into the quadratic equation, and the axis of symmetry and minimum point were calculated using standard formulas for quadratic functions. ⚹◊What is the formula to find the axis of symmetry for the quadratic function given by $f(x) = ax^2 + bx + c$?⍭ Generate me a similar question◊

Question

It is given that the graph of quadratic function f(x)=2x²+ kx+7 passes through the point (5,17).
(i) Calculate the value of k.
(ii) Find the equation of the axis of symmetry of the graph of quadratic
function.
(iii) Find the coordinates of the minimum point of the graph of the quadratic
function.

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ To find the value of $ k $, we substitute the point $ (5, 17) $ into the quadratic function $ f(x) = 2x^2 + kx + 7 $: $ 2(5)^2 + k(5) + 7 = 17 $. ‖ ‖ step 2 ⋮ Simplifying the equation: $ 2(25) + 5k + 7 = 17 $ leads to $ 50 + 5k + 7 = 17 $. ‖ ‖ step 3 ⋮ Combining like terms gives $ 5k + 57 = 17 $. Subtracting 57 from both sides results in $ 5k = 17 - 57 $, which simplifies to $ 5k = -40 $. ‖ ‖ step 4 ⋮ Dividing both sides by 5 yields $ k = -8 $. ‖ ‖ step 5 ⋮ The equation of the axis of symmetry for a quadratic function $ ax^2 + bx + c $ is given by $ x = -\frac{b}{2a} $. Here, $ a = 2 $ and $ b = -8 $. ‖ ‖ step 6 ⋮ Substituting the values gives $ x = -\frac{-8}{2(2)} = \frac{8}{4} = 2 $. ‖ ‖ step 7 ⋮ To find the coordinates of the minimum point, we substitute $ x = 2 $ back into the function $ f(x) = 2x^2 + kx + 7 $: $ f(2) = 2(2)^2 + (-8)(2) + 7 $. ‖ ‖ step 8 ⋮ Simplifying gives $ f(2) = 2(4) - 16 + 7 = 8 - 16 + 7 = -1 $. Thus, the minimum point is $ (2, -1) $. ‖ ∻Answer∻ ⚹ (i) $ k = -8 $, (ii) Axis of symmetry: $ x = 2 $, (iii) Minimum point: $ (2, -1) $ ⚹ ∻Key Concept∻ ⚹ The axis of symmetry and minimum point of a quadratic function are derived from its coefficients. ⚹ ∻Explanation∻ ⚹ The value of $ k $ was found by substituting the given point into the quadratic equation, and the axis of symmetry and minimum point were calculated using standard formulas for quadratic functions. ⚹◊What is the formula to find the axis of symmetry for the quadratic function given by $f(x) = ax^2 + bx + c$?⍭ Generate me a similar question◊

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