QUESTION IMAGE
Question
match each function with its graphical representation.
you have 0 of 18 cards correct.
$x=1+\sqrt{13}$
$2x^2 + 3x - 5 = 4$
$-x^2 + 3x = -8$
$\frac{1}{2}x^2 + 2x + 8 = 5$
$x = -3$
$-4x^2 = 24x + 11$
$x = \frac{3 \pm \sqrt{41}}{2}$
$x = \frac{3}{2}$ or $x = -3$
$x = -\frac{1}{2}$ or $x = -5\frac{1}{2}$
$\frac{1}{2}x^2 - 6x = 2$
$x = 6 \pm 2\sqrt{10}$
$x^2 - 2x = 12$
To solve this problem of matching each quadratic function (or equation) with its graphical representation, we analyze the key features of each quadratic (parabola) such as direction (opening up/down), vertex, roots (x - intercepts), and solve the equations to find roots for comparison. Here's a breakdown of a few examples:
1. Analyze \( -x^2 + 3x = -8 \) (rewrite as \( -x^2 + 3x + 8 = 0 \) or \( x^2 - 3x - 8 = 0 \))
- Direction: The coefficient of \( x^2 \) is \( -1 \) (negative), so the parabola opens downward.
- Roots: Use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) for \( ax^2 + bx + c = 0 \). Here, \( a = 1 \), \( b = -3 \), \( c = -8 \):
\( x = \frac{3 \pm \sqrt{(-3)^2 - 4(1)(-8)}}{2(1)} = \frac{3 \pm \sqrt{9 + 32}}{2} = \frac{3 \pm \sqrt{41}}{2} \).
This matches the solution \( x = \frac{3 \pm \sqrt{41}}{2} \). Look for a downward - opening parabola with these roots.
2. Analyze \( 2x^2 + 3x - 5 = 4 \) (rewrite as \( 2x^2 + 3x - 9 = 0 \))
- Factor: \( 2x^2 + 3x - 9 = (2x - 3)(x + 3) = 0 \).
- Roots: \( 2x - 3 = 0 \implies x=\frac{3}{2} \); \( x + 3 = 0 \implies x = - 3 \).
This matches the solution \( x=\frac{3}{2} \) or \( x = - 3 \). Look for a parabola (opening up, since \( a = 2>0 \)) with x - intercepts at \( \frac{3}{2} \) and \( -3 \).
3. Analyze \( \frac{1}{2}x^2 + 2x + 8 = 5 \) (rewrite as \( \frac{1}{2}x^2 + 2x + 3 = 0 \) or multiply by 2: \( x^2 + 4x + 6 = 0 \))
- Discriminant: \( b^2 - 4ac=(4)^2 - 4(1)(6)=16 - 24=-8<0 \).
- Conclusion: No real roots (parabola does not cross the x - axis). The coefficient of \( x^2 \) is \( \frac{1}{2}>0 \), so it opens upward. Look for an upward - opening parabola with no x - intercepts.
4. Analyze \( \frac{1}{2}x^2 - 6x = 2 \) (rewrite as \( \frac{1}{2}x^2 - 6x - 2 = 0 \) or multiply by 2: \( x^2 - 12x - 4 = 0 \))
- Roots: Use the quadratic formula \( x=\frac{12\pm\sqrt{(-12)^2 - 4(1)(-4)}}{2(1)}=\frac{12\pm\sqrt{144 + 16}}{2}=\frac{12\pm\sqrt{160}}{2}=\frac{12\pm4\sqrt{10}}{2}=6\pm2\sqrt{10} \).
This matches the solution \( x=\frac{6\pm2\sqrt{10}}{2}=6\pm2\sqrt{10} \) (simplified). The parabola opens upward (\( a=\frac{1}{2}>0 \)).
5. Analyze \( x^2 - 2x = 12 \) (rewrite as \( x^2 - 2x - 12 = 0 \))
- Roots: Use the quadratic formula \( x=\frac{2\pm\sqrt{(-2)^2 - 4(1)(-12)}}{2(1)}=\frac{2\pm\sqrt{4 + 48}}{2}=\frac{2\pm\sqrt{52}}{2}=\frac{2\pm2\sqrt{13}}{2}=1\pm\sqrt{13} \).
This matches the solution \( x = 1\pm\sqrt{13} \). The parabola opens upward (\( a = 1>0 \)).
6. Analyze \( -4x^2 = 24x + 11 \) (rewrite as \( 4x^2+24x + 11 = 0 \))
- Factor: \( 4x^2+24x + 11=(2x + 1)(2x + 11)=0 \).
- Roots: \( 2x+1 = 0\implies x =-\frac{1}{2} \); \( 2x + 11 = 0\implies x=-\frac{11}{2}=-5\frac{1}{2} \).
This matches the solution \( x =-\frac{1}{2} \) or \( x=-5\frac{1}{2} \). The parabola opens downward (\( a=-4<0 \)).
To fully match all functions to their graphs, repeat this process:
- Identify the direction of the parabola (sign of \( a \) in \( y = ax^2+bx + c \)).
- Find the roots (x - intercepts) by solving the quadratic equation (factoring, quadratic formula, or completing the square).
- Match the parabola’s shape (direction) and x - intercepts to the given graphs.
For example, if we consider the equation \( x^2 - 2x = 12 \) (which simplifies to \( x = 1\pm\sqrt{13} \)) and its corresponding graph:
- The parabola opens upward (since \( a = 1>0 \)).
- The roots are \( 1+\sqrt{13}\approx1 + 3.605 = 4.605 \) and \( 1-\sqrt{13}\approx1 - 3.605=-2.605 \).
Find the upward - opening par…
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To solve this problem of matching each quadratic function (or equation) with its graphical representation, we analyze the key features of each quadratic (parabola) such as direction (opening up/down), vertex, roots (x - intercepts), and solve the equations to find roots for comparison. Here's a breakdown of a few examples:
1. Analyze \( -x^2 + 3x = -8 \) (rewrite as \( -x^2 + 3x + 8 = 0 \) or \( x^2 - 3x - 8 = 0 \))
- Direction: The coefficient of \( x^2 \) is \( -1 \) (negative), so the parabola opens downward.
- Roots: Use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) for \( ax^2 + bx + c = 0 \). Here, \( a = 1 \), \( b = -3 \), \( c = -8 \):
\( x = \frac{3 \pm \sqrt{(-3)^2 - 4(1)(-8)}}{2(1)} = \frac{3 \pm \sqrt{9 + 32}}{2} = \frac{3 \pm \sqrt{41}}{2} \).
This matches the solution \( x = \frac{3 \pm \sqrt{41}}{2} \). Look for a downward - opening parabola with these roots.
2. Analyze \( 2x^2 + 3x - 5 = 4 \) (rewrite as \( 2x^2 + 3x - 9 = 0 \))
- Factor: \( 2x^2 + 3x - 9 = (2x - 3)(x + 3) = 0 \).
- Roots: \( 2x - 3 = 0 \implies x=\frac{3}{2} \); \( x + 3 = 0 \implies x = - 3 \).
This matches the solution \( x=\frac{3}{2} \) or \( x = - 3 \). Look for a parabola (opening up, since \( a = 2>0 \)) with x - intercepts at \( \frac{3}{2} \) and \( -3 \).
3. Analyze \( \frac{1}{2}x^2 + 2x + 8 = 5 \) (rewrite as \( \frac{1}{2}x^2 + 2x + 3 = 0 \) or multiply by 2: \( x^2 + 4x + 6 = 0 \))
- Discriminant: \( b^2 - 4ac=(4)^2 - 4(1)(6)=16 - 24=-8<0 \).
- Conclusion: No real roots (parabola does not cross the x - axis). The coefficient of \( x^2 \) is \( \frac{1}{2}>0 \), so it opens upward. Look for an upward - opening parabola with no x - intercepts.
4. Analyze \( \frac{1}{2}x^2 - 6x = 2 \) (rewrite as \( \frac{1}{2}x^2 - 6x - 2 = 0 \) or multiply by 2: \( x^2 - 12x - 4 = 0 \))
- Roots: Use the quadratic formula \( x=\frac{12\pm\sqrt{(-12)^2 - 4(1)(-4)}}{2(1)}=\frac{12\pm\sqrt{144 + 16}}{2}=\frac{12\pm\sqrt{160}}{2}=\frac{12\pm4\sqrt{10}}{2}=6\pm2\sqrt{10} \).
This matches the solution \( x=\frac{6\pm2\sqrt{10}}{2}=6\pm2\sqrt{10} \) (simplified). The parabola opens upward (\( a=\frac{1}{2}>0 \)).
5. Analyze \( x^2 - 2x = 12 \) (rewrite as \( x^2 - 2x - 12 = 0 \))
- Roots: Use the quadratic formula \( x=\frac{2\pm\sqrt{(-2)^2 - 4(1)(-12)}}{2(1)}=\frac{2\pm\sqrt{4 + 48}}{2}=\frac{2\pm\sqrt{52}}{2}=\frac{2\pm2\sqrt{13}}{2}=1\pm\sqrt{13} \).
This matches the solution \( x = 1\pm\sqrt{13} \). The parabola opens upward (\( a = 1>0 \)).
6. Analyze \( -4x^2 = 24x + 11 \) (rewrite as \( 4x^2+24x + 11 = 0 \))
- Factor: \( 4x^2+24x + 11=(2x + 1)(2x + 11)=0 \).
- Roots: \( 2x+1 = 0\implies x =-\frac{1}{2} \); \( 2x + 11 = 0\implies x=-\frac{11}{2}=-5\frac{1}{2} \).
This matches the solution \( x =-\frac{1}{2} \) or \( x=-5\frac{1}{2} \). The parabola opens downward (\( a=-4<0 \)).
To fully match all functions to their graphs, repeat this process:
- Identify the direction of the parabola (sign of \( a \) in \( y = ax^2+bx + c \)).
- Find the roots (x - intercepts) by solving the quadratic equation (factoring, quadratic formula, or completing the square).
- Match the parabola’s shape (direction) and x - intercepts to the given graphs.
For example, if we consider the equation \( x^2 - 2x = 12 \) (which simplifies to \( x = 1\pm\sqrt{13} \)) and its corresponding graph:
- The parabola opens upward (since \( a = 1>0 \)).
- The roots are \( 1+\sqrt{13}\approx1 + 3.605 = 4.605 \) and \( 1-\sqrt{13}\approx1 - 3.605=-2.605 \).
Find the upward - opening parabola with x - intercepts near \( 4.6 \) and \( -2.6 \).
By systematically analyzing the direction of opening, roots, and other features (like vertex, y - intercept) for each quadratic, you can match each function to its correct graphical representation.