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The axis of symmetry of a quadratic function is a vertical line that divides the parabola into two mirror-image halves. It's crucial for determining the shape and position of the parabola on a graph. In mathematical terms, the axis of symmetry for a quadratic function in standard form, which is given by \( y = ax^2 + bx + c \), can be calculated with the formula:\[ x = -\frac{b}{2a} \]In this problem, we know that the axis of symmetry is given as \( x = 1 \). This information allows us to find the relationship between the coefficients \( a \) and \( b \) by substituting into the formula:\[ 1 = -\frac{b}{2a} \]This simplifies to \( b = -2a \). Understanding the axis of symmetry is key to graphing quadratic functions accurately, ensuring that both halves are perfectly symmetrical.

The y-intercept of a quadratic function is the point where the graph crosses the y-axis. In simpler terms, it tells us the value of \( y \) when \( x = 0 \). For any quadratic function written in standard form: \( y = ax^2 + bx + c \),the y-intercept is represented by the constant term \( c \). This is because when \( x = 0 \), the equation simplifies to \( y = c \).In the exercise provided, we know that the y-intercept is 1, which allows us to directly set \( c = 1 \). Recognizing and using the y-intercept effectively helps in quickly sketching the graph of the quadratic equation, providing a starting point for your parabola on the Cartesian plane. Key points about the y-intercept:

• Occurs at \( x = 0 \)
• It’s the constant term in the quadratic equation
• Helps in identifying the position of the parabola along the y-axis

An x-intercept of a quadratic function is a point where the graph crosses the x-axis. At these points, the value of \( y \) is zero. For the quadratic equation given by \( y = ax^2 + bx + c \), the x-intercepts are found by solving the equation \[ ax^2 + bx + c = 0 \]X-intercepts are critical in understanding the roots of the quadratic function. In our scenario, we have a special condition where there is only one x-intercept. This occurs when the discriminant of the quadratic equation is zero.Typically, quadratic functions can have zero, one, or two x-intercepts, which are influenced by the value of the discriminant. Knowing how many x-intercepts exist helps in determining the nature of the quadratic graph.

The discriminant in a quadratic equation provides valuable insight into the nature of the graph of the function. For a quadratic equation of the form \( ax^2 + bx + c = 0 \), the discriminant is defined as:\[ b^2 - 4ac \]The discriminant tells you about the number of real roots the quadratic equation has:

• If \( b^2 - 4ac > 0 \), there are two distinct real roots (two x-intercepts).
• If \( b^2 - 4ac = 0 \), there is exactly one real root (one x-intercept, and the vertex touches the x-axis).
• If \( b^2 - 4ac < 0 \), there are no real roots (the parabola does not touch the x-axis).

For the exercise, because we have only one x-intercept, we set the discriminant to zero:\[ (-2a)^2 - 4a \cdot 1 = 0 \]Simplified, this provides the equation \( 4a^2 - 4a = 0 \), which allows us to find the value for the coefficient \( a \). Understanding the discriminant is essential for analyzing and predicting the behavior of quadratic equations.