Andragradsfunktion och räta linjer
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9. Figuren visar grafen till en andragradsfunktion $f$.
a) Bestäm en funktion $g$, där $y = g(x)$, som uppfyller villkoren
- grafen till $g$ är en rät linje
- $g(x) \le f(x)$ för $0 \le x \le 2$
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b) Bestäm ekvationen för en rät linje som varken skär grafen till funktionen $f$ eller grafen till funktionen $g$ för något värde på $x$.
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This question includes visual content: A coordinate system showing the graph of a quadratic function $y=f(x)$. The parabola has its vertex at $(0, 4)$ and passes through points like $(2, 3)$ and $(4, 0)$. The x-axis ranges from $-2$ to $5$ and the y-axis from $-2$ to $5$. A grid is visible behind the curve. In sub-question (a), the student is asked to find a linear function $g(x)$ that is less than or equal to $f(x)$ on the interval $[0, 2]$. Sub-question (b) asks for a line that does not intersect either $f(x)$ or the student's chosen $g(x)$.
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In this problem, we are looking at the graph of a quadratic function f and need to find a linear function g that satisfies certain conditions.
Andragradsfunktion $f(x)$
For part a, we need to find a straight line g of x such that g of x is less than or equal to f of x for x values between zero and two.
a) Hitta en rät linje $g(x)$
Villkor: $g(x) \le f(x)$ för $0 \le x \le 2$
Let's identify some points on the graph of f in this interval. When x is zero, y is four. When x is two, y is three.
A simple way to find g of x is to pick a horizontal line that stays below the lowest point of f in that interval. The lowest value in that interval is three, at x equals two.
En rät linje kan vara en horisontell linje.
If we choose g of x equals zero, it will definitely be less than or equal to the values of f in this range. Or, we can choose g of x equals one.
Now for part b, we need a straight line that never intersects either f or g. Let's look at the maximum value of f.
b) En linje som aldrig skär $f(x)$ eller $g(x)$
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