Gegeven is de parabool \(f(x) = a x^{2} + 6 x + 3 \text{.}\)
2p
Voor welke \(a\) gaat \(f\) door het punt \(A (2 , -1) \text{?}\)
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\(\begin{rcases}a x^{2} + 6 x + 3 \\ \text{door } A (2 , -1)\end{rcases} \begin{matrix}a ⋅ 2^{2} + 6 ⋅ 2 + 3 = -1\end{matrix}\)
1p
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\(4 a + 15 = -1\)
\(4 a = -16\)
\(a = -4 \text{.}\)
1p
Gegeven is de parabool \(f(x) = -4 x^{2} + b x - 8 \text{.}\)
2p
Voor welke \(b\) gaat \(f\) door het punt \(A (-2 , -38) \text{?}\)
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\(\begin{rcases}-4 x^{2} + b x - 8 \\ \text{door } A (-2 , -38)\end{rcases} \begin{matrix}-4 ⋅ (-2)^{2} + b ⋅ -2 - 8 = -38\end{matrix}\)
1p
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\(-2 b - 24 = -38\)
\(-2 b = -14\)
\(b = 7 \text{.}\)
1p
Gegeven is de parabool \(f(x) = -4 x^{2} + 7 x + c \text{.}\)
2p
Voor welke \(c\) gaat \(f\) door het punt \(A (1 , -3) \text{?}\)
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\(\begin{rcases}-4 x^{2} + 7 x + c \\ \text{door } A (1 , -3)\end{rcases} \begin{matrix}-4 ⋅ 1^{2} + 7 ⋅ 1 + c = -3\end{matrix}\)
1p
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\(3 + c = -3\)
\(c = -6 \text{.}\)
1p
Gegeven is de parabool \(f(x) = -\frac{1}{4} x^{2} - 5 x + c \text{.}\)
3p
Bereken de waarde van \(c\) waarvoor geldt dat \(y_{\text{top}} = 21 \text{.}\)
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\(x_{\text{top}} = {5 \over 2 ⋅ -\frac{1}{4}} = -10\)
1p
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\(y_{\text{top}} = f(-10) = -\frac{1}{4} ⋅ (-10)^{2} - 5 ⋅ -10 + c = 21\)
1p
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\(25 + c = 21\)
\(c = -4 \text{.}\)
1p
Gegeven is de parabool \(f(x) = -\frac{1}{4} x^{2} + b x + 1 \text{.}\)
4p
Bereken de waarde van \(b\) waarvoor geldt dat \(y_{\text{top}} = 5 \text{.}\)
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\(x_{\text{top}} = {-b \over 2 ⋅ -\frac{1}{4}} = 2 b\)
1p
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\(y_{\text{top}} = f(2 b) = -\frac{1}{4} ⋅ (2 b)^{2} + b ⋅ 2 b = 5\)
1p
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\(b^{2} + 1 = 5\)
\(b^{2} = 4\)
1p
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\(b = 2 ∨ b = -2 \text{.}\)
1p
De parabool \(f(x) = a x^{2} + x + c\) gaat door de punten \((3 , -26)\) en \((4 , -46) \text{.}\)
4p
Bereken algebraïsch \(a\) en \(c \text{.}\)
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\(f(3) = a ⋅ 3^{2} + 3 + c = -26\)
\(9 a + 3 + c = -26\)
\(9 a + c = -29\)
1p
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\(f(4) = a ⋅ 4^{2} + 4 + c = -46\)
\(16 a + 4 + c = -46\)
\(16 a + c = -50\)
1p
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\(\begin{cases}9 a + c = -29 \\ 16 a + c = -50\end{cases}\)
Aftrekken geeft \(-7 a = 21 \text{,}\) dus \(a = -3 \text{.}\)
1p
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Invullen geeft \(c = -29 - 9 ⋅ -3 = -2 \text{.}\)
1p
De parabool \(f(x) = a x^{2} + b x - 2\) gaat door de punten \((-3 , 31)\) en \((2 , 16) \text{.}\)
5p
Bereken algebraïsch \(a\) en \(b \text{.}\)
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\(f(-3) = a ⋅ (-3)^{2} + b ⋅ -3 - 2 = 31\)
\(9 a - 3 b - 2 = 31\)
\(9 a - 3 b = 33\)
1p
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\(f(2) = a ⋅ 2^{2} + b ⋅ 2 - 2 = 16\)
\(4 a + 2 b - 2 = 16\)
\(4 a + 2 b = 18\)
1p
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\(\begin{cases}9 a - 3 b = 33 \\ 4 a + 2 b = 18\end{cases}\) \(\begin{vmatrix}2 \\ 3\end{vmatrix}\) geeft
1p
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\(\begin{cases}18 a - 6 b = 66 \\ 12 a + 6 b = 54\end{cases}\)
Optellen geeft \(30 a = 120 \text{,}\) dus \(a = 4 \text{.}\)
1p
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Invullen geeft \(9 ⋅ 4 - 3 b = 33\)
\(-3 b = -3\)
\(b = 1 \text{.}\)
1p