Gegeven is de parabool \(f(x) = a x^{2} - 7 x + 6 \text{.}\)
2p
Voor welke \(a\) gaat \(f\) door het punt \(A (-4 , 2) \text{?}\)
○
\(\begin{rcases}a x^{2} - 7 x + 6 \\ \text{door } A (-4 , 2)\end{rcases} \begin{matrix}a ⋅ (-4)^{2} - 7 ⋅ -4 + 6 = 2\end{matrix}\)
1p
○
\(16 a + 34 = 2\)
\(16 a = -32\)
\(a = -2 \text{.}\)
1p
Gegeven is de parabool \(f(x) = -4 x^{2} + b x + 6 \text{.}\)
2p
Voor welke \(b\) gaat \(f\) door het punt \(A (2 , -4) \text{?}\)
○
\(\begin{rcases}-4 x^{2} + b x + 6 \\ \text{door } A (2 , -4)\end{rcases} \begin{matrix}-4 ⋅ 2^{2} + b ⋅ 2 + 6 = -4\end{matrix}\)
1p
○
\(2 b - 10 = -4\)
\(2 b = 6\)
\(b = 3 \text{.}\)
1p
Gegeven is de parabool \(f(x) = -x^{2} + 3 x + c \text{.}\)
2p
Voor welke \(c\) gaat \(f\) door het punt \(A (-2 , -6) \text{?}\)
○
\(\begin{rcases}-x^{2} + 3 x + c \\ \text{door } A (-2 , -6)\end{rcases} \begin{matrix}-1 ⋅ (-2)^{2} + 3 ⋅ -2 + c = -6\end{matrix}\)
1p
○
\(-10 + c = -6\)
\(c = 4 \text{.}\)
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 \(y_{\text{top}} = 35 \text{.}\)
○
\(x_{\text{top}} = {-5 \over 2 ⋅ -\frac{1}{4}} = 10\)
1p
○
\(y_{\text{top}} = f(10) = -\frac{1}{4} ⋅ 10^{2} + 5 ⋅ 10 + c = 35\)
1p
○
\(25 + c = 35\)
\(c = 10 \text{.}\)
1p
Gegeven is de parabool \(f(x) = -\frac{1}{4} x^{2} + b x - 9 \text{.}\)
4p
Bereken de waarde van \(b\) waarvoor geldt dat \(y_{\text{top}} = -8 \text{.}\)
○
\(x_{\text{top}} = {-b \over 2 ⋅ -\frac{1}{4}} = 2 b\)
1p
○
\(y_{\text{top}} = f(2 b) = -\frac{1}{4} ⋅ (2 b)^{2} + b ⋅ 2 b - 9 = -8\)
1p
○
\(b^{2} - 9 = -8\)
\(b^{2} = 1\)
1p
○
\(b = 1 ∨ b = -1 \text{.}\)
1p
De parabool \(f(x) = a x^{2} - 3 x + c\) gaat door de punten \((2 , 7)\) en \((3 , 14) \text{.}\)
4p
Bereken algebraïsch \(a\) en \(c \text{.}\)
○
\(f(2) = a ⋅ 2^{2} - 3 ⋅ 2 + c = 7\)
\(4 a - 6 + c = 7\)
\(4 a + c = 13\)
1p
○
\(f(3) = a ⋅ 3^{2} - 3 ⋅ 3 + c = 14\)
\(9 a - 9 + c = 14\)
\(9 a + c = 23\)
1p
○
\(\begin{cases}4 a + c = 13 \\ 9 a + c = 23\end{cases}\)
Aftrekken geeft \(-5 a = -10 \text{,}\) dus \(a = 2 \text{.}\)
1p
○
Invullen geeft \(c = 13 - 4 ⋅ 2 = 5 \text{.}\)
1p
De parabool \(f(x) = a x^{2} + b x + 4\) gaat door de punten \((-3 , -26)\) en \((2 , -6) \text{.}\)
5p
Bereken algebraïsch \(a\) en \(b \text{.}\)
○
\(f(-3) = a ⋅ (-3)^{2} + b ⋅ -3 + 4 = -26\)
\(9 a - 3 b + 4 = -26\)
\(9 a - 3 b = -30\)
1p
○
\(f(2) = a ⋅ 2^{2} + b ⋅ 2 + 4 = -6\)
\(4 a + 2 b + 4 = -6\)
\(4 a + 2 b = -10\)
1p
○
\(\begin{cases}9 a - 3 b = -30 \\ 4 a + 2 b = -10\end{cases}\) \(\begin{vmatrix}2 \\ 3\end{vmatrix}\) geeft
1p
○
\(\begin{cases}18 a - 6 b = -60 \\ 12 a + 6 b = -30\end{cases}\)
Optellen geeft \(30 a = -90 \text{,}\) dus \(a = -3 \text{.}\)
1p
○
Invullen geeft \(9 ⋅ -3 - 3 b = -30\)
\(-3 b = -3\)
\(b = 1 \text{.}\)
1p