The arrow diagram shows part of the function $f: x \rightarrow a x^{2}+b x+c$.
Find
(a) the values of $a, b$ and $c$,
(b) the positive number $x$ whose image is 2 .

Question

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ We have the function $f(x) = ax^2 + bx + c$ and the mappings: $f(-2) = -3$, $f(1) = 6$, and $f(3) = 32$. We can set up the following equations: ‖ ‖ 1. $4a - 2b + c = -3$ ‖ ‖ 2. $a + b + c = 6$ ‖ ‖ 3. $9a + 3b + c = 32$ ‖ ‖ step 2 ⋮ We can solve these equations simultaneously. First, we can eliminate $c$ by subtracting the second equation from the first and third equations: ‖ ‖ From (1) - (2): $3a - 3b = -9 \implies a - b = -3$ (Equation 4) ‖ ‖ From (3) - (2): $8a + 2b = 26 \implies 4a + b = 13$ (Equation 5) ‖ ‖ step 3 ⋮ Now we can solve Equations 4 and 5 together: ‖ ‖ From Equation 4: $b = a + 3$ ‖ ‖ Substitute into Equation 5: $4a + (a + 3) = 13 \implies 5a + 3 = 13 \implies 5a = 10 \implies a = 2$ ‖ ‖ step 4 ⋮ Substitute $a = 2$ back into Equation 4 to find $b$: ‖ ‖ $b = 2 + 3 = 5$ ‖ ‖ step 5 ⋮ Finally, substitute $a = 2$ and $b = 5$ into the second equation to find $c$: ‖ ‖ $2 + 5 + c = 6 \implies c = -1$ ‖ ‖ step 6 ⋮ Now we have $a = 2$, $b = 5$, and $c = -1$. To find the positive $x$ value whose image is 2, we set $f(x) = 2$: ‖ ‖ $2x^2 + 5x - 1 = 2 \implies 2x^2 + 5x - 3 = 0$ ‖ ‖ step 7 ⋮ We can use the quadratic formula to solve for $x$: ‖ ‖ $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-5 \pm \sqrt{25 + 24}}{4} = \frac{-5 \pm 7}{4}$ ‖ ‖ This gives us $x = \frac{1}{2}$ (positive solution) and $x = -3$ (negative solution). ‖ ∻Answer∻ ⚹ a = 2, b = 5, c = -1; positive x value is $\frac{1}{2}$ ⚹ ∻Key Concept∻ ⚹ Finding coefficients of a quadratic function using given mappings and solving for specific function values. ⚹ ∻Explanation∻ ⚹ We determined the coefficients $a$, $b$, and $c$ by setting up equations based on the function's mappings and then solved for the positive $x$ value that results in an output of 2. ⚹◊What is the significance of the coefficients $a$, $b$, and $c$ in the quadratic function $f(x) = ax^2 + bx + c$?⍭ Generate me a similar question◊

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