Digital SAT Math Practice Question #1179
Test your problem-solving accuracy on this official-caliber item. Select an answer choice to check your reasoning instantly.
When a line is tangent to a parabola, how many points of intersection do they share?
If you set the equations of the line and parabola equal to each other, what mathematical condition must be true for the resulting quadratic equation to have exactly one solution?
⚡ **Calculus Shortcut:** For students familiar with derivatives, this problem can be solved very quickly. 1. The slope of the tangent line to the parabola \( y = x^2 - 4x + 7 \) is given by its derivative: \( y' = 2x - 4 \). 2. The given line \( y = 2x + k \) has a slope of 2. 3. For tangency, the slope of the parabola at the point of tangency must equal the slope of the line: \( 2x - 4 = 2 \). 4. Solve for \( x \): \( 2x = 6 \implies x = 3 \). This is the x-coordinate of the point of tangency. 5. Find the y-coordinate by plugging \( x=3 \) into the parabola's equation: \( y = (3)^2 - 4(3) + 7 = 9 - 12 + 7 = 4 \). 6. Since the point of tangency \( (3, 4) \) also lies on the line \( y = 2x + k \), substitute these values into the line's equation: \( 4 = 2(3) + k \). 7. Solve for \( k \): \( 4 = 6 + k \implies k = -2 \). **Graphing Calculator/Desmos:** Graph the parabola \( y = x^2 - 4x + 7 \). Then, graph the line \( y = 2x + k \) and use a slider for \( k \). Adjust the slider until the line just touches the parabola at a single point. The value of \( k \) at that point will be the answer.
To find the value of 'k' for which the line is tangent to the parabola, we need to determine when the system of equations has exactly one solution. This occurs when the discriminant of the resulting quadratic equation is zero. 1. **Set the ...
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