Coëfficiënten in kwadratische formules
Gegeven is de parabool \(f(x) = a x^{2} + 5 x - 3 \text{.}\)
2p
Voor welke \(a\) gaat \(f\) door het punt \(A (4 , -15) \text{?}\)
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\(\begin{rcases}a x^{2} + 5 x - 3 \\ \text{door } A (4 , -15)\end{rcases} \begin{matrix}a ⋅ 4^{2} + 5 ⋅ 4 - 3 = -15\end{matrix}\)
1p
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\(16 a + 17 = -15\)
\(16 a = -32\)
\(a = -2 \text{.}\)
1p
Gegeven is de parabool \(f(x) = -3 x^{2} + b x - 5 \text{.}\)
2p
Voor welke \(b\) gaat \(f\) door het punt \(A (-4 , -49) \text{?}\)
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\(\begin{rcases}-3 x^{2} + b x - 5 \\ \text{door } A (-4 , -49)\end{rcases} \begin{matrix}-3 ⋅ (-4)^{2} + b ⋅ -4 - 5 = -49\end{matrix}\)
1p
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\(-4 b - 53 = -49\)
\(-4 b = 4\)
\(b = -1 \text{.}\)
1p
Gegeven is de parabool \(f(x) = -2 x^{2} + 8 x + c \text{.}\)
2p
Voor welke \(c\) gaat \(f\) door het punt \(A (1 , 11) \text{?}\)
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\(\begin{rcases}-2 x^{2} + 8 x + c \\ \text{door } A (1 , 11)\end{rcases} \begin{matrix}-2 ⋅ 1^{2} + 8 ⋅ 1 + c = 11\end{matrix}\)
1p
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\(6 + c = 11\)
\(c = 5 \text{.}\)
1p
Gegeven is de parabool \(f(x) = -\frac{2}{5} x^{2} + 4 x + c \text{.}\)
3p
Bereken de waarde van \(c\) waarvoor geldt dat \(y_{\text{top}} = 2 \text{.}\)
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\(x_{\text{top}} = {-4 \over 2 ⋅ -\frac{2}{5}} = 5\)
1p
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\(y_{\text{top}} = f(5) = -\frac{2}{5} ⋅ 5^{2} + 4 ⋅ 5 + c = 2\)
1p
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\(10 + c = 2\)
\(c = -8 \text{.}\)
1p
Gegeven is de parabool \(f(x) = \frac{1}{4} x^{2} + b x - 6 \text{.}\)
4p
Bereken de waarde van \(b\) waarvoor geldt dat \(y_{\text{top}} = -15 \text{.}\)
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\(x_{\text{top}} = {-b \over 2 ⋅ \frac{1}{4}} = -2 b\)
1p
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\(y_{\text{top}} = f(-2 b) = \frac{1}{4} ⋅ (-2 b)^{2} + b ⋅ -2 b - 6 = -15\)
1p
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\(-b^{2} - 6 = -15\)
\(b^{2} = 9\)
1p
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\(b = 3 ∨ b = -3 \text{.}\)
1p
De parabool \(f(x) = a x^{2} + 3 x + c\) gaat door de punten \((3 , -14)\) en \((4 , -25) \text{.}\)
4p
Bereken algebraïsch \(a\) en \(c \text{.}\)
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\(f(3) = a ⋅ 3^{2} + 3 ⋅ 3 + c = -14\)
\(9 a + 9 + c = -14\)
\(9 a + c = -23\)
1p
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\(f(4) = a ⋅ 4^{2} + 3 ⋅ 4 + c = -25\)
\(16 a + 12 + c = -25\)
\(16 a + c = -37\)
1p
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\(\begin{cases}9 a + c = -23 \\ 16 a + c = -37\end{cases}\)
Aftrekken geeft \(-7 a = 14 \text{,}\) dus \(a = -2 \text{.}\)
1p
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Invullen geeft \(c = -23 - 9 ⋅ -2 = -5 \text{.}\)
1p
De parabool \(f(x) = a x^{2} + b x - 5\) gaat door de punten \((2 , 3)\) en \((4 , 35) \text{.}\)
5p
Bereken algebraïsch \(a\) en \(b \text{.}\)
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\(f(2) = a ⋅ 2^{2} + b ⋅ 2 - 5 = 3\)
\(4 a + 2 b - 5 = 3\)
\(4 a + 2 b = 8\)
1p
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\(f(4) = a ⋅ 4^{2} + b ⋅ 4 - 5 = 35\)
\(16 a + 4 b - 5 = 35\)
\(16 a + 4 b = 40\)
1p
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\(\begin{cases}4 a + 2 b = 8 \\ 16 a + 4 b = 40\end{cases}\) \(\begin{vmatrix}2 \\ 1\end{vmatrix}\) geeft
1p
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\(\begin{cases}8 a + 4 b = 16 \\ 16 a + 4 b = 40\end{cases}\)
Aftrekken geeft \(-8 a = -24 \text{,}\) dus \(a = 3 \text{.}\)
1p
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Invullen geeft \(4 ⋅ 3 + 2 b = 8\)
\(2 b = -4\)
\(b = -2 \text{.}\)
1p