Coëfficiënten in kwadratische formules
Gegeven is de parabool \(f(x) = a x^{2} - 3 x + 7 \text{.}\)
2p
Voor welke \(a\) gaat \(f\) door het punt \(A (4 , -21) \text{?}\)
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\(\begin{rcases}a x^{2} - 3 x + 7 \\ \text{door } A (4 , -21)\end{rcases} \begin{matrix}a ⋅ 4^{2} - 3 ⋅ 4 + 7 = -21\end{matrix}\)
1p
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\(16 a - 5 = -21\)
\(16 a = -16\)
\(a = -1 \text{.}\)
1p
Gegeven is de parabool \(f(x) = x^{2} + b x - 8 \text{.}\)
2p
Voor welke \(b\) gaat \(f\) door het punt \(A (4 , -4) \text{?}\)
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\(\begin{rcases}x^{2} + b x - 8 \\ \text{door } A (4 , -4)\end{rcases} \begin{matrix}4^{2} + b ⋅ 4 - 8 = -4\end{matrix}\)
1p
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\(4 b + 8 = -4\)
\(4 b = -12\)
\(b = -3 \text{.}\)
1p
Gegeven is de parabool \(f(x) = -4 x^{2} - 6 x + c \text{.}\)
2p
Voor welke \(c\) gaat \(f\) door het punt \(A (2 , -27) \text{?}\)
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\(\begin{rcases}-4 x^{2} - 6 x + c \\ \text{door } A (2 , -27)\end{rcases} \begin{matrix}-4 ⋅ 2^{2} - 6 ⋅ 2 + c = -27\end{matrix}\)
1p
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\(-28 + c = -27\)
\(c = 1 \text{.}\)
1p
Gegeven is de parabool \(f(x) = \frac{2}{3} x^{2} - 4 x + c \text{.}\)
3p
Bereken de waarde van \(c\) waarvoor geldt dat \(y_{\text{top}} = 2 \text{.}\)
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\(x_{\text{top}} = {4 \over 2 ⋅ \frac{2}{3}} = 3\)
1p
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\(y_{\text{top}} = f(3) = \frac{2}{3} ⋅ 3^{2} - 4 ⋅ 3 + c = 2\)
1p
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\(-6 + c = 2\)
\(c = 8 \text{.}\)
1p
Gegeven is de parabool \(f(x) = -\frac{1}{6} x^{2} + b x + 4 \text{.}\)
4p
Bereken de waarde van \(b\) waarvoor geldt dat \(y_{\text{top}} = 10 \text{.}\)
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\(x_{\text{top}} = {-b \over 2 ⋅ -\frac{1}{6}} = 3 b\)
1p
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\(y_{\text{top}} = f(3 b) = -\frac{1}{6} ⋅ (3 b)^{2} + b ⋅ 3 b + 4 = 10\)
1p
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\(1\frac{1}{2} b^{2} + 4 = 10\)
\(b^{2} = 4\)
1p
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\(b = 2 ∨ b = -2 \text{.}\)
1p
De parabool \(f(x) = a x^{2} - x + c\) gaat door de punten \((-2 , -16)\) en \((3 , -41) \text{.}\)
4p
Bereken algebraïsch \(a\) en \(c \text{.}\)
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\(f(-2) = a ⋅ (-2)^{2} - 1 ⋅ -2 + c = -16\)
\(4 a + 2 + c = -16\)
\(4 a + c = -18\)
1p
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\(f(3) = a ⋅ 3^{2} - 1 ⋅ 3 + c = -41\)
\(9 a - 3 + c = -41\)
\(9 a + c = -38\)
1p
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\(\begin{cases}4 a + c = -18 \\ 9 a + c = -38\end{cases}\)
Aftrekken geeft \(-5 a = 20 \text{,}\) dus \(a = -4 \text{.}\)
1p
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Invullen geeft \(c = -18 - 4 ⋅ -4 = -2 \text{.}\)
1p
De parabool \(f(x) = a x^{2} + b x + 3\) gaat door de punten \((3 , 9)\) en \((4 , 19) \text{.}\)
5p
Bereken algebraïsch \(a\) en \(b \text{.}\)
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\(f(3) = a ⋅ 3^{2} + b ⋅ 3 + 3 = 9\)
\(9 a + 3 b + 3 = 9\)
\(9 a + 3 b = 6\)
1p
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\(f(4) = a ⋅ 4^{2} + b ⋅ 4 + 3 = 19\)
\(16 a + 4 b + 3 = 19\)
\(16 a + 4 b = 16\)
1p
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\(\begin{cases}9 a + 3 b = 6 \\ 16 a + 4 b = 16\end{cases}\) \(\begin{vmatrix}4 \\ 3\end{vmatrix}\) geeft
1p
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\(\begin{cases}36 a + 12 b = 24 \\ 48 a + 12 b = 48\end{cases}\)
Aftrekken geeft \(-12 a = -24 \text{,}\) dus \(a = 2 \text{.}\)
1p
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Invullen geeft \(9 ⋅ 2 + 3 b = 6\)
\(3 b = -12\)
\(b = -4 \text{.}\)
1p